overview · web viewfunctional approach. 65. travers, kenneth j. algebra 2 with trigonometry....

Advanced Mathematics: Unit II- Advancing with Polynomials and ExponentsChapter 1: Working with Polynomial Functions

Overview

Chapter I expand the students’ ability to use polynomials to represent and solve problems reflecting real-world situations while focusing on symbolic and graphical patterns.

The study of the properties and graphs of polynomial functions is useful to scientists, astronomers, physicists and chemists in the field of scientific research. These properties are useful in making satellite dishes, car headlights, radio telescopes and reflecting telescopes.

The ideas and skills learned in this chapter will help the students organize information, interpret and solve problems logically. They will also enable the students to come up critical evaluations of the solutions found.

Throughout the chapter, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a common pattern as elaborated upon in the instructional procedure.

Classroom activities are designed to actively engage students in problem investigation and making sense of problem situations. They will work together collaboratively in heterogeneous groupings: in pairs or in small groups. They will communicate their mathematical thinking and the results of their group efforts.

Focus Questions

1. How can polynomial equations be used to provide accurate models of practical problems that involve three dimensions?

2. How can a polynomial model be used to solve problems where maxima or minima are of significant importance?

3. How can polynomial expressions be used to represent and predict social or fiscal changes over time?

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Objectives

BEC Standards

After completing Chapter 1, the students should be able to:

identify a polynomial function from a given set of relations;

determine the degree of a given polynomial function;

find the quotient of polynomials by using the division algorithm and synthetic division;

find the quotient using synthetic division and the Remainder Theorem when p(x) is divided by (x-c);

state and illustrate the Remainder Theorem;

find the value of p(x) for x = k by synthetic division and the Remainder Theorem;

state and illustrate the Factor Theorem;

find the zeroes of polynomial functions of degree greater than two by using:

Factor Theorem Factoring Synthetic Division Depressed Equations

draw the graph of polynomial functions of degree greater than two (use a graphing calculator, if available).

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Key TermsTerms when numbers are added or subtracted, they are called

terms. Example: 4x² + 7x − 8 is a sum of three terms.

Factors when numbers are multiplied, they are called factors. Example: (x + 1)(x + 2)(x + 3) is a product of three factors.

Variable is a symbol that takes on values.

Value is a number; thus if x is the variable and has the value 4, then 5x + 1 has the value 21.

Constant is a symbol that has a single value. Example: The symbols '5' and ' ' are constants.

The beginning letters of the alphabet a, b, c, etc. are typically used to denote constants, while the letters x, y, z, are typically used to denote variables.

Example: if we write y = ax² + bx + c, we mean that a, b, c are constants (i.e. fixed numbers), and that x and y are variables.

Monomial in x is a single term of the form axn, where a is a real number and n is a whole number. Examples: 5x3, −6.3x, 2.

Polynomial in x is a sum of monomials in x. Example: 5x3 − 4x² + 7x − 8

Degree of a Term is the sum of the exponents of all the variables in that term. In functions of a single variable, the degree of a term is simply the exponent. Example: The term 5x³ is of degree 3 in the variable x.

Leading Term of a Polynomial is the term of highest degree. Example: The leading term of this polynomial 5x³ − 4x² + 7x − 8 is 5x³.

Leading Coefficient of a Polynomial is the coefficient of the leading term. Example: the leading coefficient of that polynomial is 5.

Degree of a Polynomial is the degree of the leading term. Example: the degree of this polynomial 5x³ − 4x² + 7x − 8 is 3.

Constant Term of a Polynomial is the term of degree 0; it is the term in which the variable does not appear. Example: The constant term of this polynomial 5x³ − 4x² + 7x − 8 is −8.

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General Form of a Polynomial shows the terms of all possible degree. Example, is the general form of a polynomial of the third degree: ax³ + bx² + cx + d. Notice that there are four constants: a, b, c, d.

Polynomial function has the form: y = A polynomial

A polynomial function of the first degree, such as y = 2x + 1, is called a linear function; while a polynomial function of the second degree, such as y = x² + 3x − 2, is called a quadratic.

Domain and range The natural domain of any polynomial function is: − < x < . x may take on any real value on the x-axis.

The domain of a function is the set of values of the independent variable, which are the values of x.

The range of a function is the set of values of the dependent variable, which are the values of y.

Polynomial equation is a polynomial set equal to 0. P(x) = 0.

Example. P(x) = 5x³ − 4x² + 7x − 8 = 0

Root, or Zero, of a polynomial is a solution to the polynomial equation,

P(x) = 0. That is, the number r is a root of a polynomial P(x) if and only if P(r) = 0.

Remainder Theorem if a polynomial P(x) is divided by a linear function x - c, the remainder is P(c). The degree of the remainder is always one less than the degree of the divisor.

Factor Theorem If x − c is a factor of a polynomial P(x), then P(x) = 0.

Synthetic division is a shortcut method of doing long division of polynomials when the divisor is of the form x + c.

Rational Root Theorem - In the polynomial, an xn + an-1 xn-1 + · · · + a1 x + a0 = 0 where the coefficients an ,an-1 , . . . ,a1 ,a0 are integers.

If a rational number c/d, is factored to its lowest terms, is a solution to the equation, then c is a factor of the constant term of the polynomial and d is a factor of the leading coefficient of the polynomial.

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http://www.themathpage.com/aPreCalc/functions.htm#domain


Concept Map

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Algebraic Expressions

POLYNOMIALS

Evaluation of polynomials

by substitution

Operations on polynomials

Terms and degrees of

polynomials

Addition

Subtraction

Multiplication

Division

Polynomial functionsP(x) = AnXn+An-1Xn-1+…+ A1X+A0

Algorithm

Remainder Theorem

Factor Theorem

can be simplified through

Graphing of polynomial functions

Zeroes of polynomial functions

Rational zeroes

Polynomial inequalities

leads to

can be illustrated by

are performed and expressed using

in the form of

are explainedthrough


Lesson 1 RECOGNIZING POLYNOMIALS

TIME

1 session

SETTING

Computer room and Math room

OBJECTIVES

At the end of this lesson, the students should be able to:

tell whether an expression is a polynomial or a non-polynomial function;

identify polynomials from a given number of algebraic expressions; and

state the characteristics of polynomials.

PREREQUISITE

Students should have learned the following concepts and their definitions:

1. algebraic expressions 5. monomial

2. term 6. binomial

3. coefficient 7. trinomial

4. polynomials 8. degree of polynomial

RESOURCES

chart

Manila paper

marker pen

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http://www.clipart.com/en/close-up?o=756019&memlevel=C&a=c&q=cliparts%20in%20math&s=31&e=60&show=&c=&cid=&findincat=&g=&cc=&page=2%00%E5%A1%B9%EF%92%81%E1%B4%BB%E4%A1%BF%E2%B2%AF%E5%B6%82%E8%97%84%E6%8C%A7%00%00%EA%AE%A5

http://www.clipart.com/en/close-up?o=1374250&memlevel=A&a=c&q=cliparts%20in%20math&s=1&e=30&show=&c=&cid=&findincat=&g=&cc=&page=%00%E5%A1%B9%EF%92%81%E1%B4%BB%E4%A1%BF%E2%B2%AF%E5%B6%82%E8%97%84%E6%8C%A7%00%00%EA%AE%A5

http://www.clipart.com/en/close-up?o=954962&memlevel=B&a=c&q=cliparts%20in%20math&s=31&e=60&show=&c=&cid=&findincat=&g=&cc=&page=2%00%E5%A1%B9%EF%92%81%E1%B4%BB%E4%A1%BF%E2%B2%AF%E5%B6%82%E8%97%84%E6%8C%A7%00%00%EA%AE%A5


PROCEDURE

Opening Activity: POLYNOMIAL QUIZ BEE

A. Begin with a review of algebraic expressions. Ask volunteers to form two groups. Acting as contestants, they will answer the three questions asked by the teacher. For the first set, the contestants will translate the following mathematical expressions to phrases.

1. 3x + 4 (Answer: 4 is added to thrice a number)2. 6x2 (Answer: a number, squared, is multiplied by 6)

3. 3. (Answer: a number is multiplied by 2/3, or twice a

number is divided by 3)

B. For the second set of questions, students will translate the following English phrases to mathematical expressions.

1. A number x is raised to the 2/3 power. (Answer: x2/3)

2. Six times a number x added to the product of three and the cube of (Answer: 6x + 3c3)

3. The square root of x subtracted from the product of three and x. (Answer: 3x - )

4. Three divided by the product of 4 and X. Answer:

C. For the third set of questions, students will express the following using nonnegative exponents.

a. a-2

b. ab-1

c. 2x-4

d. a2b-3

D. Process the activity by asking:

Why do we have to translate mathematical symbols to English and vice versa?

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It is easier to solve word problems if we know how to take the English problem and translate it into mathematical statements which are also known as algebraic expressions.

How can algebraic expressions be used in understanding polynomials?

A polynomial is basically an algebraic expression that consists of a sum of terms, each term being the product of a constant and a nonnegative (or zero) power of a variable or variables. For example: 3x3 - 2x + 0.5x2 + 6

Main Activity

A. Given the students’ basic understanding of polynomials, ask them to identify the polynomials in the following questions:

1. What is the difference between and ?

2. Which of the two is a polynomial?

B. Ask the students to give examples of mathematical expressions and let them determine whether the given examples are polynomials or non-polynomials.

Discussion Ideas

Considering the translated mathematical symbols in our review and given the set of algebraic expressions below, ask the students to answer the following:

Group I Group IIa.) 6x + 3c2 a.) 3x-3

b.) 6x2 b.) 6x1/2

c.) 3x + 4 c.) 3x +

d.) d.)

1. What can be said about the exponents of the variables in the first group of examples? The exponents in the second group of examples?

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2. Compare example b in both groups. How do they differ?

3. Look at the second term in example c in both groups. How are they different from each other?

4. Where are the variables in example d in both groups located?

5. What characteristics of the terms in Group I make them polynomials?

6. Are all polynomials algebraic expressions? Why or why not?

7. Are all algebraic expressions polynomials? Why or why not?

Summarize the responses and formulate the key learning points on polynomials.

Key Learning Points

A polynomial is an algebraic expression involving only nonnegative-integer powers of one or more variables and containing no variable in the denominator.

Polynomial functions are functions with x as an input variable. They are made up of several terms, each of which consists of two factors, the first being a real number coefficient, and the second being x (or any variable) raised to some nonnegative integer power.

Polynomial functions are functions of the form: f(x) = anxn + an-

1xn-1 + ... + a1x + a0.

The value of n must be a nonnegative integer. That is, it must be a whole number equal to zero or a positive integer.

The numerical coefficients of the terms in the polynomial function are an, an-1,..., a1, a0. They are real numbers.

The degree of the polynomial function is the highest value for n where an is not equal to 0.

Extension Ideas

1) Which of the following are polynomials? Give the reason for your answer.

a.) 2x3 d.)

b.) 7 – 3a2 e.) 5x2 – 2xy + 3y2

c.) x3 + x-2 f.) 3x +

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(Answers: a, b, e are polynomials. The rest have variables with negative or non-integer powers)

2) Based on the definition given previously, why is the expression

not a polynomial?

(Answer: It is not a polynomial because is equal to x–2, which is

a negative power of the variable x.)

Closing Activity

Ask the students to use the words “term” and “degree” in two sentences each. In one sentence, the mathematical meaning of the word should be used. In the other sentence, the non-mathematical meaning of the word should be used.

Example:

The term 5x means five multiplied by x; meanwhile, x3 means x to the third power.

ASSESSMENT

Which of the following expressions is a polynomial? Write P before the number if the expression is a polynomial and NP if it is a non-polynomial.

1. 4x5 – 3x4 + 2x3 – x2 + 2x5 + 6

2. 6x + 2x + 5 + 3x

3. 2/3 x7 – x2 + 8

4. 10x2 + 3x – 4x – 7

5. 3x4 + x1/3 + 1

HOMEWORK

State whether each of the given expressions is a polynomial. Write P if the expression is a polynomial and NP if it is a non-polynomial.

1. 6x + 3x3 6. x-2

2. x1/2 7. x/2

3. x 8. 5x - 3

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4. 3x/4 9. 8 – 5b2

5. 2/4x 10. 12x4

REFERENCES

Dalton, Leroy C. Using Algebra

Foster, Alan G. Algebra 2 With Trigonometry. Applications and Connections.

Jose-Dilao, Soledad. Advanced Algebra, Trigonometry and Statistics.

Functional Approach. 65.

Travers, Kenneth J. Algebra 2 with Trigonometry.

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Lesson 2 TERMS AND DEGREES OF A POLYNOMIAL

TIME

1 session

SETTING

Math room

OBJECTIVES


classify polynomials according to their number of terms; and

determine the degree of the polynomial.

PREREQUISITE

Students are expected to have understood the concept of polynomials. They should have imbibed the skill of identifying polynomials from a list of algebraic expressions.

RESOURCES

drill board or flash cards

chart

PROCEDUREOpening Activity

A. Conduct a review on polynomials by asking: When is an expression a polynomial?

B. Ask the students to present and discuss their assignments in Lesson 1. As an alternative activity, test the students’ skill in identifying polynomials. Using flash cards, ask the students to tell whether each of the given algebraic expressions is a polynomial or not.

1. 6x + 3x3 5.

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2. x1/2 6. x-2

3. x 7. y2 - 5

4. 8.

C. Introduce the new lesson using a motivational activity. Introduce the lesson by telling the students that:

It is common practice that names like MACMAC, JUNJUN, and JETJET are written thus: MAC2, JUN2, JET2. Mathematically speaking, MAC2 is MACC, where only C is taken twice. What do (MAC)2 and MAC2 represent?

(Possible Answer: The above examples represent the degree of the variables and the application of one of the laws of exponents. (Mac)2 is read as MMAACC while MAC2 is read as MACC.)

Main Activity

Present the lesson by giving three groups of examples. Ask the students to observe the similarities and differences between the variables and the exponents. Let them write their observations.

Group A Group B Group C

a. 6abc a. 6a + bc a. a + b + cb. 3x2y2z b. 3x2 + y2z3 b. 3x2 – 3y2 + z2

c. equ c. eq – gu c. e5 – g4 – u3

d. 5 d. 5x – xo d. 5x2 - 1

Discussion Ideas

A. Discuss the concepts “terms” and “degrees of polynomials” by asking the following questions:

1. Compare the examples given in each group. How do they differ?

2. How many terms are given in each of the following?

a.) group A

b.) group B

c.) group C

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3. How would you classify the examples in

a.) group A?

b.) group B?

c.) group C?

4. When is a polynomial

a.) a monomial?

b.) a binomial?

c.) a trinomial?

If a polynomial has four or more terms, how is it classified?

5. In group A, what is the highest exponent of the variables? In group B? In group C?

6. In example c group C, 5 is the highest exponent, therefore the polynomial is of degree 5. How is the degree of a polynomial determined?

Extension Ideas

Allow the students to work on the following exercises in order to practice their skills in identifying the terms and degrees of polynomials.

A. Identify each polynomial according to the number of its terms.

1. x2 + 10 x + 5

2. 3a + 2b

3. 4xyz

4. 2m2n-mn2

5. 7a + b -2

B. Give the degree of each of the following polynomials

1. 3x2 –x2 + x -3

2. x2 – x6 + x4 + 3

3. 5x2y – 4x4y3-2

4. x + 2x2 + 3x2 + 6

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5. 3x6 + 6x4 + x2 - x

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Closing Activity

Summarize the key learning points by asking each group to complete the following phrases:

1. The degree of a term is……..

2. The degree of a polynomial is…….

3. The standard form of a polynomial is arranged in……

4. Polynomials are classified into….

5. The degree of a nonzero constant polynomial is….

6. The zero polynomial does not have any…..

Key Answers1. the sum of the exponents of the variable.

2. the highest of the degrees of its terms (after it has been simplified)

3. descending order

4. monomials, binomials, trinomials and multinomials

5. zero (0)

6. degree

ASSESSMENT

Arrange the following polynomials in standard form. Classify each polynomial according to the number of its terms. Then determine the degree of each of the following polynomials:

1. 4x2 – x2 + x – 2

2. 3x2 – x6 + x4 + 1

3. 5x2 – x6 – 4x4

4. 5x – 2x2 + 3x2 + 6

5. 6x6 + 6x4 + x2 – x

6. 2x2 + 8x – x10 - 5 + 20x5

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HOMEWORK

Rewrite each of the following polynomials in descending powers of the variable and find the degree of each polynomial:

1. 2 6. 2 – ½ t – 2t2

2. -5 – 2x 7. x – 7x3 + x4 – 3x2

3. x4 + 2 – 3x 8. m2 – 11 - m4. ½ m – ¼ m2 9. 6y + 7y2 – 2 – 5y3

5. 5 – 2p + 6p2 10. 13m2

REFERENCES

Cruz, F.R. Expanding Mathematics.

Jose-Dilao. Advanced Algebra, Trigonometry and Statistics. Functional Approach. 65–66.

Math IV: Advanced Algebra, Trigonometry and Statistics – BEC

Travers, Dalton and Brunner. Using Algebra. Third edition.

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Lesson 3EVALUATING POLYNOMIALS EXPRESSIONS

TIME

1 session

SETTING

Math room

OBJECTIVE

At the end of this lesson, the students should be able to evaluate polynomial expressions for specified values of the variables.

PREREQUISITEStudents are expected to know the following concepts or to have the following skills:

determining the degree of polynomials; and

classification of polynomials according to the number of terms.

RESOURCES

drill board or flash cards

chart


a. Recall the concepts learned the previous day. Discuss the assigned items. Focus on items that were not accurately done by the students.

b. Using flash cards, prepare two samples per mathematical operation.

c. Invite the students to evaluate a set of integers using the four fundamental operations on integers.

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d. Sample exercises on integer operations:

Set A

e. After the short drill, explain to the students that they will apply their basic knowledge on how to add, subtract, multiply and divide integers in evaluating polynomial expressions.

Main Activity

1. Relate the concept of polynomials with a simple machine. Ask the students:

Do you know what pencils are made of?

What materials are needed in the manufacture of pencils? Materials that are needed to produce an output are called inputs.

If mathematical symbols/expressions constitute inputs, what would the output be if we put these symbols in a simple machine?

2. Post on the board a simple machine which will be used to evaluate the following expressions.

What is -6 + 19 ?

What is 8 + (-7) ?

What is -15 + 16 ?

What is 17 + (-3) ?

What is -5 + 9 ?

What is -4 - (16) ?

What is 12 - (-4) ?

What is -18 - (6) ?

What is -6 - (-10)?

What is -8 - (-16)? What is -3 x 3 ?

What is 7 x 7 ?

What is 1 x 7 ?

What is 8 x 3 ?

What is 9 x 9 ?

What is -21 ÷7) ?

What is 40 ÷(-4) ?

What is -42 ÷(-7)?

What is -6 ÷ (6) ?

What is 5 ÷ (1) ?

Set B

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http://www.mathguide.com/lessons/#divide

http://www.mathguide.com/lessons/#multiply

http://www.mathguide.com/lessons/#subtract

http://www.mathguide.com/lessons/#add


Example 1: Evaluate 3x when x = 2

3x

output

Suppose the following values of x were used as inputs in the machine. What would the corresponding output be?

when x = 3 3x = 3(3) = 9 x = -2 3x = 3(-2) = -6

x = -5 3x = 3(-5) = -15

Example 2: If f(x) = -x3 + 2x2 + x – 3, find the value of the function when x = 2.

SolutionF(2) = -(2)3 + 2(2)2 + (2) – 3

= -8 + 2(4) + 2 – 3 = -8 + 8 + 2 – 3 = -1

Therefore, the value of the polynomial function when x = 2 is –1.

3. Divide the class into 5 groups. Give each group a strip of Manila paper where each of the following is written. Let the groups find the value of f(x) and tell them to write their solutions on the board.

a. f(x) = 2x2 + 3x – 5 where x = -1

b. f(x) = -4x3 – x4 + 3x + 1 x = 1

c. f(x) = x3 – 2x2 + 3 x = 4

d. f(x) = -3x2 + 5x x = -2

e. f(x) = -2x + 26 x = 0

This machine multiplies every input(x) by 3

x = 2

6

input

The value of the polynomial function is 6.

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Discussion Ideas

Ask the students to answer the following questions:

1. How is a polynomial function f evaluated for a given value of x?

2. What are the steps in evaluating a polynomial function?

3. How are polynomials added or subtracted?

4. How are polynomials multiplied and divided? What laws should be followed?

Extension IdeasSeatwork

Ask the students to simplify the expressions below and find their values when x = -2 and y = 3

a. 4x + 2y + 3(2x-5y)

b. 4x2 – 5xy2 + 3xy (2x-3y)

Closing Activity

Summarize the key learning points as follows:

To find the value of a polynomial function, substitute the given value of x and evaluate the resulting numerical expression.

The value of a polynomial function is the value of the

polynomial for a given value of the variable.

ASSESSMENT

Evaluate the following, given that x = -1, y = 2, z = 3.

1. x2 + xy + y2

2. 2x2 + 3y – z

3.

4. 4x4 + 5y3 + y2 + z3

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HOMEWORK

Evaluate the given polynomials for the specified values of the variables.

a. -3x2 + 3 for x = -5

b. 4x2 – y for x = 2 and y = -3

c. xy2z – 5 for x = 7, y = 3 and z = -5

REFERENCES

Addison and Wesley. Algebra and Trigonometry.

Cruz. Expanding Mathematics.


Travers. Algebra 2 With Trigonometry.

Yu-hico. Experiencing Mathematics IV for High School

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Lesson 4 ADDITION AND SUBTRACTION OF

POLYNOMIALS

TIME

1 session

SETTING

Math room

OBJECTIVES


add and subtract polynomial expressions, and

apply some formulas in adding and subtracting polynomial expressions.

PREREQUISITE

1. Students should have already learned how to evaluate polynomial expressions.

2. Students should know the different properties of multiplication, addition, etc.

RESOURCES

drill board or flash cards chart


1. Ask the students to recall the concepts learned the previous day and to discuss the assigned items.

Distributive Property

+ (a – b); –(a – b), the Rule of Signs

How do you simplify grouping symbol/s preceded by a plus (+)sign? a minus (-) sign?

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2. Introduce the fundamental operations on polynomials. Ask the students:

What methods are used in adding and subtracting polynomial expressions?

3. Present some illustrative examples for students to analyze until they have identified the two methods, namely, the horizontal and the vertical methods.

Example 1: Add 3x2 + 2 and 7x2 – 6x + 3

Solution

Horizontal Method Vertical Method(3x2 + 2) + (7x2 – 6x + 3) 3x2 + 2

7x2 – 6x + 3

10x2 – 6x + 5

10x2 – 6x + 5

Example 2: Simplify (8x2 – 6x + 7) – (4x2 + 3x – 12)

Solution

Horizontal Method Vertical Method (8x2 – 6x + 7) – (4x2 + 3x – 12) 8x2 – 6x + 7

-4x – 3x + 12 4x2– 9x + 19

4x2 – 9x + 19

4. Allow individual students to do some practice exercises. Ask them to use the two methods in adding and subtracting the following polynomial expressions.

a) (7x2 – 3y2) + (7x2 + 3y2)

b) (-3n2 + n – 8) + ( 6n2 – 2n – 5)

c) (12y2 + 4y + 7) – (9y2 – 8y – 6)

d) (2m2n2 – 6mn – 13) – (6m2n2 – 3mn – 7)

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5. Provide a lecture-discussion on polynomials in the real world. See Teacher Notes.

Main Activity

Divide the class into 5 groups. Give each group a strip of Manila paper wherein all the necessary solutions should be recorded. Instruct the students to solve the problem accurately by applying the formula given to them.

Problem 1Find the surface area of a metal cylinder if r = 12 cm and h = 25 cm. Use 3.14 for the value of pi.

Problem 2A room has a perimeter of 44 feet and a height of 8 feet. The room has three 4 feet by 4 feet windows and one 2 feet by 6 feet door. Find the number of rolls of 2 feet by 5 feet wallpaper needed to cover up the computer room.

Discussion Ideas

Ask the students to present their work on the board and to assign one member per group to discuss the results of their activity.

Extension Ideas

Challenge the students to find the force on an aquarium window that is 2 feet below the water surface if the window is 5 feet wide by 4 feet high. Use the formula below.

F = 31h2w + 62dhw

Where: d = depth, in feet, of a window below the surface of water

h = height in feet

w = width in feet

Note: F is the force in pounds.

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Closing Activity

Synthesize the important points of the lesson by asking the students to verify if the following statements are true:

To add two polynomials, place a plus sign between them, then simplify. To simplify, combine all like terms. Then, if possible, arrange the terms in descending powers of a variable.

The subtraction property, a – b = a + (-b), can be used to subtract one polynomial from another. In subtracting polynomial expressions, simply follow the rules on the subtraction of integers by changing the sign of the subtrahend then proceed to addition.

Distributive Property

ASSESSMENT

Find the sum or difference of the following polynomials:

a. 5x3y – 3 (2x3y + 5x) + 18x – 3b. 5x + 3 (2x + 4) – 5 (2y + 3) + 18yc. 3x(2x + 4) – 2x2 + 5 (x – 3)d. 16xy2z + 4x2yz – (8x2yz + 2xy2z)

HOMEWORK

1) Find the surface area of a metal cylinder whose radius is 18 cm and whose height is 32 cm.

2) The computer room has a perimeter of 56 feet and a height of 12 feet. The room has two 4 feet by 5 feet windows and a 3 feet by 6 feet door. Find the number of rolls of 2 by 5 feet wallpaper needed to cover up the computer room.

REFERENCES



Dalton, Leroy C. Using Algebra. Third Edition.

Yu-hico. Experiencing Mathematics IV For High Schools.

27





Teacher NotesPOLYNOMIALS ON THE JOB

Formulas are used in different fields of study. Often the right side of a formula is a polynomial wherein the addition and subtraction of polynomials are usually involved. Some formulas in the operation of polynomial expressions have connections to real-life situations.

Example 1)

A metalworker wants to find the surface area S of a metal container shaped like a cylinder. (The teacher will show to the class the illustration of a cylinder).

The formula for the surface area of a cylinder is:

S = 2 (3.14) r2 + 2 (3.14) rh,

where: r is the radius of the base

h is the height,

is approximated by 3.14

Example 2)

Wallpaper hangers often use formulas to estimate how much wallpaper is needed for a room.

Formula: R = ph – (½) n

where: r is the number of rolls of wallpaper needed

p is the perimeter of a room in feet

h is the height of the walls in feet

n is the number of normal-sized doors and windows.

28



Lesson 5 MULTIPLICATION OF POLYNOMIALS

TIME

One session

SETTING

Math room

OBJECTIVE

At the end of this lesson, the students should be able to find the product of two polynomial expressions.

PREREQUISITE

Students should have knowledge of the laws of multiplication of exponents.a.) xa . xb = xa + b b.) ( xa)b = xab

c.) ( xayc)b = xab.ybc

d.) x0 = 1, x 0

RESOURCES

drill board or flash cards chart Manila paper marker pen

PROCEDURE

Opening Activity

A. Have a review on the addition and subtraction of polynomial expressions and the application of some formulas in the real world.

29







B. Introduce the FOIL method for multiplying polynomial expressions to the students. The FOIL method is an application of the distributive property that makes the multiplication of polynomials faster.The product of two binomials is the sum of the products of the:

irst terms

uter terms

nner terms

ast terms

C. lllustrate how to multiply binomials using the FOIL Method:

Example 1 Use the FOIL method to find (9a – 3) (a + 4)

Solution: (9a – 3)(a + 4)= 9a2 + 36a – 3a – 12= 9a2 + 33a - 12

Example 2 Find (2x2 + 10x – 4)(x – 12)

Solution: (2x2 + 10x – 4)( x – 12)= (2x2 + 10x – 4)x – (2x2 + 10x – 4)12 (Distributive

Property)= 2x3 + 10x2 – 4x – 24x2 – 120x + 48= 2x3 – 14x2 – 124x + 48

Main Activity: MULTIPLICATION OF POLYNOMIALS

Divide the class into 5 groups. Give each group one whole Manila paper and a marker pen. Distribute the Activity Sheets, one for each group.

Instruction: Find each product.

1. (a + b)(a2 – ab + b2) 6. (2x – 3)(x2 – 3x – 8)

2. (x – y)(x2 + xy + y3) 7. (m – 4)(3m2 + 5m – 4)

3. r(r – 2)(r – 3) 8. (b + 1)(b – 2)(b + 3)

4. (2x – 3)(x + 1)(3x – 2) 9. (2a + 1)(a – 2)2

5. (a – b)(a2 + ab + b2) 10. (2k + 3)(k2 – 7k + 21)

30


Discussion Ideas

1. Ask the students to present their work on the board and assign one member per group to discuss the results of the activity.

2. Ask the students: What strategies or steps were applied by your group in multiplying polynomials?

3. Summarize their responses, then reinforce their ideas by presenting the following key learning points:

In the multiplication of polynomials, three important properties of exponents are useful, namely:

Product of PowersLet x be any real number, and let a and b be any positive integers. Then (xa)(xb) = xa+b

Power of a PowerLet x be any real number, and let a and b be any positive integers. Then (xa)b = xab

Power of a ProductLet x and y be any real numbers, and let a and b be any positive integers. Then (xy)a = xaya

The Distributive Property and the FOIL method can be used in multiplying polynomial expressions.

In multiplying a polynomial by another polynomial, the vertical and horizontal methods of finding products are also applicable.

Closing Activity

Ask the students to work in pairs. Let each one give some tips on how to multiply polynomials the easy way. Let them share their ideas with their respective partners.

31



ASSESSMENT

Find each product.

1. (3x + 2)(5x + 1)

2. (4xy + 5)(3xy – 6)

3. (y – 3z)(y2 + 3yz + 9z2)

4. (x + 1)(x – 1)(x + 1)

5. (x2 + 4xy + 4y2)(x + 2y)

HOMEWORK

Find the area of each figure.

1. (k2 – 7k + 21) 2.

(2k + 3) (2a + 4b)

3. 4. (5x2 – 6x + 10)

(4x – 6) (2x + 4)

REFERENCES



Dalton, Leroy C. Using Algebra. Third edition.

Yu-hico. Experiencing Mathematics IV for High School.

32




Student Activity 5MULTIPLYING POLYNOMIALS

Challenge Problem

Show geometrically that (a + b)2 = a2 + 2ab + b2

Solution

a. Draw the diagram of a square that has sides of length a + b.

a b

b b

a a

a b

b. The area of the square can be expressed as (a + b)2 or by a2 + 2ab + b2.

33


Lesson 6 DIVISION OF POLYNOMIALS

TIME

One session

SETTING

Math room

OBJECTIVE

At the end of this lesson, the students should be able to find the quotient when a polynomial in x is divided by x – c, using long division.

PREREQUISITE

Students should have gained knowledge of the multiplication of polynomial expressions by applying the three properties of exponents.

RESOURCES drill board or flash cards chart Manila paper marker pen

PROCEDURE

Opening Activity

A. Using flashcards, drill the students on the four fundamental operations on integers.

Review them on the multiplication of polynomials using the FOIL Method and the Distributive Property. Likewise, check the students’ assignments.

34







B. Begin the new lesson with a short activity named “Number Puzzle”.

Group the students by threes and let them answer the cross number puzzle. Allow them to use a calculator so that they can solve the puzzle in 2 to 3 minutes. Answers should always be ready for posting.

Across Down1. 122 1. 2(3 + 4 – 10 + 12)

5. 8[(3 + 2) + (4 + 1)] 2. (2)(2)(2)(5)

6. (9 – 2)(9 + 2) 4. (8 + 9)(9 – 8)

7. 9(2) 6. 7(10) + 8(1)

8. (20 + 3)(10 + 7) 7. (12 + 5)(12 – 5)

11. (5 + 2)(11 – 4) 9. 9(6 + 4) + 2(2)

10. (2 + 6)2

The Number Puzzle

Answer Key

1 2 3 4

5 3 6

1 2 7 4

8 9 10

1 1

1 4 4 1

8 0 3 7 7

1 2 1 8 4

3 9 1 6

1 4 9 4

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Main Activity: DIVIDING POLYNOMIALS

Demonstrate the division process. Ask the students to observe the long division process for real numbers. Let them compare long division of polynomials with the division process for real numbers.

a) Divide 4325 by 20

Solution: 20

b) Divide x2 – 2x + 1 by x – 1

Solution: x - 1

A remainder of 0 indicates that the divisor is a factor of the dividend. To check the division, remember the Division Algorithm:

Dividend = (Divisor)(Quotient) + Remainder

x2 – 2x + 1 = (x – 1) (x – 1) + 0

c) Divide x4 + 3x + 2 by x2 + 1

3240

12520

5120

r. 5

36


Solution:

x2 + 1

-x2 + 3x + 2-x 2 - 1 3x + 3


x4 + 3x + 2 = (x2 + 1)(x2 – 1) + (3x + 3)

Guide Questions

In example a, how many terms are given in the dividend? In the divisor? How about in example b? In example c?

a) What did you notice about the arrangement of terms in the dividend in the second and third examples? Why is it necessary to arrange the terms in descending order?

b) How is the first term of the quotient obtained? What would be the next step?

c) What operations are involved in order to have the next dividend?

Divide the class into 5 groups that will work on a challenging problem. Distribute the Activity Sheets. Assign one member of the group to report the results of its work.

Extension Ideas

1. Ask the students if they know some applications of dividing polynomials in the real world.

2. Let them work on a sample problem related to vaccination.

Yu Kamin is a genetic engineer working on a vaccine for influenza. The number of people in a small town who catch influenza during an epidemic is estimated to be:

N = where: N represents the number of people

37


t represents the number of weeks from the start of the epidemic.

One week after an epidemic starts, 170(1)2/[12 + 1] or 85 people would have influenza. What if the epidemic lasts for a long time? What will happen to the total population of the small town?

Complete the table below and show all the necessary computations. What happens to the value of N as t increases?

Total Number of Weeks

Number of People Affected

1.5 weeks3 weeks

7.5 weeks12 weeks18 weeks21 weeks27 weeks

Closing Activity

1. Emphasize the key learning points.

The long division process used for real numbers can also be used in dividing polynomials. A remainder of 0 indicates that the divisor is a factor of the dividend.

The division algorithm states that:


2. Ask the students to arrange the basic steps in dividing a polynomial by another polynomial from 1 to 7.

Write the divisor (x – c) on the left-hand side of the dividend

Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.

38



Write the given expression in descending order of the exponents. Consider 0 the coefficient of any missing power of x.

Subtract the product from the dividend to obtain the new set of terms in the dividend. Repeat this procedure until you reach the last term of the quotient.

Check your answer by multiplying the quotient and the divisor. The result should be the same as the dividend.

Multiply the quotient by the two terms in the divisor.

Answer Key

1. Write the given expression in descending order of the exponents. Consider 0 the coefficient of any missing power of x.

2. Write the divisor (x – c) on the left-hand side of the dividend

3. Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.

4. Multiply the quotient by the two terms in the divisor.

5. Subtract the product from the dividend to obtain the new set of terms in the dividend. Repeat this procedure until you reach the last term of the quotient.

6. Check your answer by multiplying the quotient and the divisor. The result should be the same as the dividend.

ASSESSMENT

Divide using long division.

1. (12x2 + 8x – 15) by (2x + 3)

2. (6x3 + x2 – 4x + 1) by (3x – 1)

3. (2y3 + 5y2 – y – 1) by (2y – 1)

4. (2m3 - 5m2 + 3) by (m2 – 2m – 4)

5. (3x2 – 2x – 4) by (x – 3)

39



HOMEWORK

Answer the following problems:

1. Divide 9b2 + 9b – 10 by 3b – 2

2. Simplify (5m2 – 34m – 7)(m – 7)

3. Show that 4 – n is a factor of n3 – 6n2 + 13n – 20

4. Simplify the formula for the number of people who catch influenza

during an epidemic, n =

Solution

t2 + 1

REFERENCES


Dalton, Leroy C. Using Algebra. Third Edition.


Yu-hico. Experiencing Mathematics IV for High Schools.

40



Student Activity 6DIVIDING POLYNOMIALS

Challenge Problem

The area of a rectangle is represented by 6x2 + 38x + 56. Its width is represented by 2x + 8. Point B is the midpoint of AC. ABFG is a square. Find the length of rectangle ACED and the area of square ABFG.

Solution

Divide 6x2 + 38x + 56 by 2x + 8 to find the length of the rectangle.

2x + 8

The length of rectangle ACED is 3x + 7. The length of one side of square ABFG is 2x + 8/2 or x + 4. The area of the square is (x + 4)2 or x2 + 8x + 16

The Figure A G D

B F 2x + 8

C E

41


Lesson 7SYNTHETIC DIVISION

TIMEOne session

SETTING

Math room

OBJECTIVE

At the end of this lesson, the students should be able to find the quotient and the remainder when a polynomial in x is divided by x – c, using synthetic division.

PREREQUISITE

Students are expected to have learned the following concepts:

Division of polynomials using long division.

Division of powers.

“Let x be any nonzero real number, and let a and b be any positive integers.”

If a = b, then xa/xb = 1If a > b, then xa/xb = xa-b

If a < b, then xa/xb = 1/xb - a

RESOURCES

drill board or flash cards chart

PROCEDURE

Opening Activity

A. To check the students’ understanding of the previous lesson, give some practice exercises on division of polynomials.

42







Simplify each of the following:

1. 24x7/12x3

2. 12x4y2/4x2y5

3. –42x5y4/6xy3

4. –4(x – y)/20(y – x)4

B. Review the students’ assignment and ask them to solve a new set of exercises on division of polynomials using the long method.

1. (12x2 + 8x – 15) by (x + 3)

2. (x2 + 8x + 16 ) by (x + 4)

3. (x3 + x2 – x – 1) by (x – 1)

4. (2x4 + 5x3 + 11x + 6) by (x + 3)

C. Ask the students to write their solutions on the board using long division. Process the activity by asking these questions:

What is the value of learning the long division method for polynomials?

Would you like to know a shorter way of dividing polynomials?

Main Activity

Present an illustrative example to the class:

Problem: What are the quotient and the remainder when (3x3 + x2 – 8x – 4) is divided by (x – 2)?

Solution

X – 2

r8

6x – 4

43


To check, multiply the quotient by the divisor and add the remainder.

(x – 2)(3x2 + 7x + 6) = 3x3 + 7x2 + 6x -6x 2 – 14x – 12

3x3 + x2 – 8x – 12

(3x3 + x2 – 8x – 12) + 8 = = 3x3 + x2 - 8x – 4

we write:

Emphasize the fact that since we are dealing with rational expressions, we cannot simply add the remainder, as we do with real numbers. We have to be careful with our notation.

Discussion Ideas

A. Discuss with the students the long division method. Try to explain how to eliminate all variables and write the problem in a more compact form. Let the students compare the long division method used above with the process below. Explain this in detail.

2 3 1 -8 -4 6 14 12

3 7 6 8 or 3x2 + 7x + 6 r8

Show why x – 2 is written as 2 in synthetic division. This is because

x – 2 means x – (2) or x = 2.

B. Present new sets of exercises and ask the students to solve them using synthetic division.

1. Divide (2x3 – 9x2 + 13x – 12) by (x – 3)

Solution:

3 2 -9 13 -12 6 -9 12

2 -3 4 0 or 2x2 – 3x + 4 r 0

Remainder is 8Divisor is x - 2

Note: Emphasize that they should always place zero for all the missing terms of the polynomial.

44


2. Divide (x5 – 4x3 + 5x2 – 5) by (x + 1)

Solution:

-1 1 0 -4 5 0 5 -1 1 3 -8 8

1 -1 -3 8 -8 3

or: x4 – x3 – 3x2 + 8x – 8 r3

3. Find the quotient and the remainder using synthetic division.

6x3 + x2 - 4x + 1 by 3x – 1

Solution

Note 3x – 1 = 3

This means that we can write the above expression as:

2

2 1 -1 0

Thus,

(divide the expression by 3 so we can use synthetic division)

45


Check:(2x2 + x – 1)(3x – 1) = 6x3 + 3x2 – 3x

-2x 2 – x + 1 6x3 + x2 – 4x + 1

Guide Questions

1. Which part of the binomial divisor serves as the divisor in this method?

2. What is done with the divisor and the numerical coefficient of the term with the highest degree?

3. Where is the result written?

4. What operations are involved in getting the next numbers?

5. How is the last number obtained? What does a remainder of zero mean?

6. What is the degree of the polynomial quotient?

7. Are there any restrictions in the use of synthetic division? When can we use this method and when is this NOT applicable?

Extension Ideas

Get the meaning of the word “synthetic” from the dictionary. In your own words, write a paragraph explaining why the process discussed earlier is called the “synthetic division process.”

Closing Activity

a) Explain the key learning points in using synthetic division as an alternative method for dividing polynomials.

Synthetic division is a division process for polynomials in one variable when the divisor is of the form x – c, where c is any real number.

Synthetic division is useful in the following cases:

a) finding the quotient and the remainder when a polynomial in x is divided by x – c; and

b) determining if a binomial of the form x – c is a factor of the given polynomial in x.

46



b) Recalling the steps used in dividing polynomials by synthetic division, ask the students to complete the following:

Steps in Synthetic Division:

1. Write the numerical coefficient of the dividend in ___________order of the exponents with 0 as the coefficient of any missing power of x.

2. Write the___________ of the divisor x – c at the left hand corner of the coefficients.

3. Bring down the ___________of the dividend, multiply it by c and add the result to the second column.

4. Multiply the ______obtained in the previous step by c and add the result to the third column. Repeat this process until you reach the last column of the dividend.

5. The third row of numbers represents the coefficient of the terms in the quotient. The degree of the variable is one less than that of the dividend and the rightmost number is the ________.

Answer Key

1. descending

2. constant term c

3. leading coefficient

4. sum

5. remainder

ASSESSMENT

Find the quotient and the remainder using synthetic division.

a) x3 – 2x2 + 4x + 1 by x – 2

b) 4x3 - 6x2 + 2x + 1 by x – ½

c) 2x4 + 5x3 + 11x + 6 by x + 3

47



HOMEWORK

Find the quotient and the remainder using synthetic division.

1. x4 + 8x2 – 5x3 – 2 + 15 x by x – 3

2. x5 – 32 by x + 2

3. 2x3 – 7x2 – 8x + 16 by x – 4

4. 3y3 + 2y2 – 32y + 2 by y – 3

5. 76x3 – 19x2 + x + 6 by x – 3

REFERENCES

Foster and Gordon. Algebra 2 With Trigonometry—Applications and Connections


Yu-hico. Experiencing Mathematics 4.

48



Lesson 8THE REMAINDER THEOREM

TIMEOne session

SETTING

Math room

OBJECTIVES

At the end of the lesson, the students should be able to:

state the Remainder Theorem; and

use synthetic division and the Remainder Theorem to find the remainder when a polynomial in x is divided by a binomial of the form x – c.

PREREQUISITE

Students should have already learned and applied the steps used in dividing polynomial expressions by means of synthetic and long division.

RESOURCES

Manila paper marker pen

PROCEDURE

Opening Activity: GUESS-A-REMAINDER

a) Have a drill to check on the students’ basic knowledge on division of whole numbers.

b) Give exercises on how to use synthetic division to find the quotient and the remainder.

1. (2x3 + 5x2 - 4x – 3) by (x + 2)

2. (2a4 + 5a3 + 7a2 – 4a + 6) by (a + 3)

49







c) Start the new lesson with a simple game following the procedure below:

1. Ask the students to give a 3-digit number.

2. Tell them to divide the number by 9.

Guide Questions

Can you find the remainder without dividing? How?

When will a remainder be equal to zero?

Main Activity: REMAINDER THEOREM

1. Develop the concept of the Remainder Theorem by presenting sample equations. Write and solve these expressions on the board.

a. 48/7 = (6 + r6) 48 = 7(6) + r6

b. 17/5 = (3 + r2) 17 = 5(3) + r2

Ask the students:

a. Can you write these equations another way?48 = 7(6) + r617 = 5(3) + r2

b. In the new equation, what name is given to each term?


2. Allow the students to solve for the quotient and the remainder using synthetic division or long division first. Then, ask them to compare their answers with P(1) in #2.1 and P(-1) in # 2.2

2.1 3x + 5 x – 1

P(x) = 3x + 5 (1) = 3(1) + 5

= 8

Therefore, the remainder is 8.

50


2.2 2x 2 + 4x + 5 x + 1

P(x) = 2x2 + 4x + 5 P(-1) = 2(-1)2 + 4(-1) + 5 = 2 + 1

= 3 Therefore, the remainder is 3.

3. Find the remainder using the Remainder Theorem:

(9a3 + 3a + 6a2 + 9) divided by (3a – 3)

Solution

(find the remainder)

Can be written as:

= (divide the expression by

3 so that the divisor is of the form a – c)

By the Remainder Theorem:

3(1)3 + 1 +2(1)2 + 3 = 3 +1 + 2 + 3 = 9

Multiply this by 3, since 3a – 3 = 3(a – 1)

The remainder when 9a3 + 3a + 6a2 + 9 is divided by 3a – 3 is 9(3) = 27

Check:

- 9a 3 – 9a 2 _______ 15a2 + 3a -15a 2 + 15a___

18a + 9 -18a +18

27

3a2 + 5a + 6

51


4. Divide the students into small groups. Have them work on the given set of problems provided in the attached Student Activity Sheet.

Discussion Ideas

1. What is the relation between the remainder and the value of the polynomial at x = c when the polynomial in x is divided by a binomial of the form x – c?

2. How is the remainder obtained when the polynomial in x is divided by a binomial of the form x –c?

3. What happens if the remainder is zero?

Note: The value of the polynomial at x = c is equal to the remainder when the divisor is x – c.

Closing Activity

a) Emphasize the key learning points:

Division Algorithm for Polynomials

For each polynomial P(x) of a positive degree, and for any number c, there exist unique polynomials Q(x) and R(x) such that:

P(x) =[ (x – c) Q(x)] + R(x),

Where: Q(x) is of degree n – 1

R(x) is the remainder

Remainder Theorem

If a polynomial P(x) is divided by x – c, then the remainder is P(c).

52



Proof of the Remainder Theorem

1. P(x) = (x – c) Q(x) + R(x) Division Algorithm for Polynomials

2. P(x) = (x – c) Q(x) + R Definition of Division of Polynomials

R(x) must have a degree less than the

degree of (x – c).

` Thus, R(x) = R is a constant (which may or may

not be 0)

3. P(c) = [(c – c) Q( c )] + R Equation (2) is true for all x. Therefore x = c

4. P(c) = 0 x Q ( c ) + R Multiplication Property

5. P(c) = R Identity

Hence, the remainder R is equal to P(c).

b) Ask the students to relate synthetic division with the Remainder Theorem in terms of evaluating polynomials.

Answer Key

Synthetic division can be used as a convenient way to find the values of polynomial functions P(c). Hand in hand with this process is the Remainder Theorem. The remainder r obtained in synthetic division is indeed equal to P(c).

ASSESSMENT

Find the remainder using the Remainder Theorem.

1. (2y3 – 5y2 – 8y – 50) divided by (y – 5)

2. (3y3 + 2y2 – y + 5) divided by (y + 2)

3. (x3 + 4x – 7) divided by (x – 3)

4. (8x4 + 2x + 4x3 + 1) divided by (x + ½)

53



HOMEWORK

Using the Remainder Theorem, find the remainder of the following:

1. (12x4 + 11x – x2 – 40x3 –90) by (x – 3)

2. (2x4 – 4x3 + 9x2 + 2x – 5) by (x – 1)

3. (4y3 – 6y2 + 2y + 1) by (2y – 1)

4. (y4 + 8y2 – 5y3 – 2 + 15y) by (y – 3)

5. (3m3 – 2m2 + 2m – 3) by (m + 3)

REFERENCES

Foster, Gordon. Algebra 2 With Trigonometry—Applications and Connections.



54



Student Activity 8

REMAINDER THEOREM APPLICATIONS

Solve the following polynomial functions:

A. Use the Remainder Theorem to find the value of x4 – 4x2 + 12x – 9

when x =2.

B. Use the Remainder Theorem to find the value of x4 – 4x3 + 12x – 9

when x = 1.

C. Determine whether x – 3 is a factor of 2x3 – 3x2 – 12x + 9.

If x – 3 is a factor, name the other factor of the polynomial.

D. Which of the following binomials are factors of x4 – 3x2 + 6x - 4?

a.) x + 2

b.) x – 1

c.) x + 3

55


Lesson 9THE VALUES OF POLYNOMIAL FUNCTIONS

TIMEOne session

SETTINGThis lesson must be done inside the math room. This is simply a continuation of the past two lessons on synthetic division and the Remainder Theorem. If the class learned the past lessons quickly, this lesson may be integrated into those lessons as no new concepts are presented here.

OBJECTIVES

At the end of the lesson, the students should be able to further develop their skills in determining the:

quotient of a polynomial divided by a binomial of the form (x - c) using synthetic division; and

value of a polynomial using synthetic division and the Remainder Theorem.

PREREQUISITE

Students should have already learned the Remainder Theorem and synthetic division.

RESOURCES

Manila paper marker pen


A. Check the students’ assignments and review the class on evaluating polynomial functions by means of a short contest:

56







B. Form two teams to evaluate polynomials for integral values of the variable. One team will use direct substitution while the other team will use synthetic division. Ask the students to work on the following practice set.

Task: Determine the remainder in each of the following polynomial expressions using synthetic division.

a.) y3 – 4y2 – 2y + 5; y – 1

b.) 2x3 – 3x2 + 2x – 8; x – 2

c.) 4m3 + 4m2 + m + 3; m – 4

Main Activity

A. Give the following set of questions for students to solve as a group. The outputs of each group will be presented to the class.

1. Given the polynomial 8x3 + 6x + 7, find its degree and leading coefficient.

(Answer: Since 3 is the highest exponent, the polynomial is of degree 3 and 8 is its leading coefficient.)

2. To find the quotient and the remainder when 3x3 – 5x + 10 is divided by x – 2, what are the initial steps you must take?

(Answer: Arrange the terms in order of descending power and write zero for the missing terms.)

3. Using long division and synthetic division, find the quotient and remainder. Compare your answers.

Long Division

x – 2

57


Synthetic Division2 3 0 - 5 10

6 12 14

3 6 7 24

3x2 + 6x + 7 24 is the remainder

3x2 + 6x + 7 is the quotient

The quotient is 3x2 + 6x + 7 and the remainder is 24.

Using either of the methods, students should be able to find the same quotient and the remainder.

5. Using the Remainder Theorem, find the value of the polynomial at x = 2.

f(x) = 3x3 – 5x + 10f(2) = 3(2)3 – 5(2) + 10

= 3(8) – 10 + 10

f(2) = 24

By the Remainder Theorem, the remainder will be equal to the value of the polynomial at x = 2.

B. Present another example:

Divide (2x3 – 4x2 – 5x + 5) by (x + 1)

Since synthetic division is much shorter, you may use this method to save time.

-1 2 -4 -5 5

-2 6 -1

2 -6 1 4

2x2 – 6x + 1 r 4

58


f(x) = 2x3 – 4x2 – 5x + 5

f(-1) = 2(-1)3 – 4(1)2 – 5(-1) + 5

= 2(-1) – 4(1) + 5 + 5

= –2 –4 + 5 + 5

= –6 + 10

f(-1) = 4

Discussion Ideas

1. When we divided 3x3 – 5x + 10 by x – 2, what did we use as divisor in the synthetic division?

2. What can you say about the remainder and the value of the polynomial?

Closing Activity

1. Synthesize the key points of the lesson by saying:

If a polynomial f(x) is divided by x – r until a remainder without x is obtained, this remainder is equal to f (r).

2. Ask students: How do you find the value of a polynomial?Answer: Apply the Remainder Theorem and synthetic division.

ASSESSMENT

Find the value of the polynomials using the Remainder Theorem and synthetic division.

1. (2y3 – 5y2 – 8y – 50) divided by (y – 5)

2. (3y3 + 2y2 – y + 5) divided by (y + 2)

3. (x3 + 4x – 7) divided by (x – 3)

4. (9a3 + 3a + 6a2 + 9) divided by (3a – 3)

5. (8x4 + 2x + 4x3 + 1) divided by (x + ½)

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HOMEWORK

Using the Remainder Theorem, find the value of the polynomials.

1. (12x4 + 11x – x2 – 40x3 –90) by (x – 3)

2. (2x4 – 4x3 + 9x2 + 2x – 5) by (x – 1)

3. (4y3 – 6y2 + 2y + 1) by (2y – 1)

4. (y4 + 8y2 – 5y3 – 2 + 15y) by (y – 3)

5. (3m3 – 2m2 + 2m – 3) by (m + 3)

REFERENCES

Foster, Gordon. Algebra 2 With Trigonometry–Applications and Connections


Yu-hico. Experiencing Mathematics 4

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Lesson 10THE FACTOR THEOREM

TIME

One session

SETTING

Math room

OBJECTIVE

At the end of this lesson, the students should be able to use the Factor Theorem and other concepts to find the zeroes of the polynomial function of degree greater than two.

PREREQUISITE

Students should have learned the zeroes of polynomial functions.

RESOURCES

Chart

PROCEDURE

Opening Activity

A. Have students practice their skills in factoring by solving the following:

4y – 24

15a3 + 20a

4x2 –81

y2 + 18y+ 81

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B. Conduct a review on finding the zeroes of a polynomial function. Present these examples as written on Manila paper for the students’ verification.

1. Given f(x) = 4x + 2, find the zero of the function.4x + 2 = 0 4x = 2 x = -2/4 or –1/2

2. If f(x) = x2 + 3x – 10, find the zeroes of the function:

x2 + 3x – 10 = 0 (x + 5)(x – 2) = 0 x = –5; x = 2

The zeroes of the function are –5 and 2.

Main Activity

A. Begin the new lesson by saying:

A linear function has only one zero, while a quadratic function, has at most two zeroes. If the degree of a given polynomial function is n, how many zeroes does the function have? How are the zeroes of a function found?

a. What is/are the zero/zeroes of the function in the previous examples?

b. How many zeroes are there in each function?

B. Develop the students’ understanding of the Factor Theorem by asking them to analyze the example below (already prepared in Manila paper for the presentation).

a) P(x) = x3 – x2 – 10x – 8

b) P(x) = x3 + x2 – x – 1

C. Discuss the Factor Theorem and relate this to the Remainder Theorem. Explain how these two theorems complement each other. Also discuss how they can be used to solve problems involving roots and the factorization of polynomials with degree higher than 2.

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The Remainder Theorem states that:

“When the polynomial P(x) is divided by x-c, the Remainder is P(c)”

The Factor Theorem

“Given the polynomial P(x). If P(c) = 0, then x-c is a factor of P(x). Conversely, if x-c is a factor of P(x), then P(x) = 0.

If the remainder when P(x) is divided by (x-c) is zero, then (x-c) is a factor of P(x).

PROOF

Suppose P(c) = 0. By the Remainder Theorem, when P(x) is divided by (x–c), the remainder R = P(c) = 0

Then, P(x) = (x-c) Q (x) + R becomes P(x) = (x–c) Q (x) + 0 P(x) = (x–c) Q (x)

Therefore (x–c) is a factor of P(x)

D. Present some illustrative examples depicting the Factor Theorem

1) Ask the students to guess the value of x in the given function: P(x) = x3 – x2 – 10x – 8

Substitute 4, -2, -1 to the given function.

If x = 4,x3 – x2 – 10x – 843 – 42–10(4) – 8 = 0(x – 4) is a factor of P(x).

If x = -2,X3 – x2 –10x – 8(-2)3 – (-2)2 – 10(-2) – 8 = 0(x + 2) is a factor of P(x)

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If x = -1

x3 – x2 – 10x – 8

(-1)3 – (-1)2 – 10(-1) – 8 = 0

(x + 1) is also a factor of P(x).

Therefore, the zeroes are 4, –2, and –1.

2) P(x) = x3 + x2 – x – 1

Substitute 1, 2, -1 to the given function.

If x = 1

x3 – x2 – x – 1

13 + (1)2 – 1 – 1

1 + 1 – 1 – 1 = 0

(x–1) is a factor of P(x)

If x = 2

x3 + x2 – x – 1

23 + 22 – 2 – 1

8 + 4 – 2 – 1

12 – 2 – 1 = 9

(x – 2) is not a factor of P(x) and 2 is not a zero of P(x)

If ax = -1

x3 + x2 – x – 1

(-1)3 + (-1)2 – (-1) – 1

-1 + 1 + 1 –1 = 0

(x + 1) is a factor of P(x).

Therefore, the only zeroes of the function are 1 and –1.

f(x)=x3 + x2 – x – 1 is a polynomial of degree 3.

We found that 1 and –1 are roots of this equation.

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Using synthetic division, we have:

1 1 1 -1 -11 2 1

-1 1 2 1 0

-1 -1

-1 1 1 0

-1

2 0

Note: From this, we can see that p(x) can be factored as:

p(x) = x3 + x2 – x – 1 = (x – 1)(x +1)(x + 1)

because –1 served as a root twice, we say –1 is a root of MULTIPLICITY 2

E. Divide the class into small groups. Have the groups work on the following sets of polynomial functions:

Which of the following binomials, (x – 1), (x – 2), (x + 2), (x –3), (x + 3) are divisors of the following expressions? Find the zeroes of their related functions.

a. x3 + 2x2 – 9x – 8

b. x4 – 4x3 – x2 + 16x – 12

c. x4 – 2x3 – 7x2 + 8x + 12

Discussion Ideas

1. What are the degrees of the functions in numbers 1 and 2?

2. How many zeroes are there in numbers 1and 2?

p(x) = x3 + x2 – x - 1

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3. How are 4, -1, -2 related to the constant term given in example number 1? How are 1, 2, -1 related to the constant term in example number 2?

4. Why is there a need to substitute these possible factors to the function?

5. How are the zeroes of a polynomial function related to the constant term of the polynomial?

6. When is a certain number a zero of a polynomial function? When is it not a zero?

7. What are the steps in finding the zeroes of P(x) using the Factor Theorem?

Closing Activity

Ask students to determine the steps in finding the zeroes of a polynomial P(x). Consolidate their responses to come up with the following steps:

1. Substitute possible values of c for x in the given polynomial P(x).

2. Use the Remainder Theorem. If P(c) = 0, then (x – c) is a factor of P(x) by virtue of the Factor Theorem.

3. Equate each factor to zero to get the zeroes of the function.

4. The number of zeroes of a function is less than or equal to the degree of the given function.

ASSESSMENT

Find the zeroes of the function:

1. P(x) = x3 + 3x2 – 2x – 6

2. P(x) = x3 – x2 – 4x + 4

3. P(x) = 3x3 – x2 + 12 – 4

HOMEWORK

Find the zeroes of the following polynomials using the Factor Theorem.

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1.) P(x) = x3 – 12x2 + 41x - 4

2.) P(x) = x4 – 3x2 + 2

REFERENCES

Foster, Gordon. Algebra 2 With Trigonometry–Applications and Connections


Mathematics IV. Advanced Algebra, Trigonometry and Statistics. 2002 BEC.


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Lesson 11ZEROES OF POLYNOMIAL FUNCTIONS: A

RECALL

TIMEOne session

SETTING

Math room

OBJECTIVES


apply the Factor Theorem, factoring techniques, synthetic and long division and the Rational Root Theorem; and

find the zeroes of polynomial functions of degree greater than 2.

PREREQUISITE

The students are expected to have learned the following concepts: Finding the zeroes of linear and quadratic functions, synthetic and

long division as well as the Remainder Theorem to solve for the quotient and the remainder of a polynomial function

RESOURCES

graph board

chart

colored chalk

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PROCEDURE

Opening Activity

A. Check the students’ assignment and review finding the quotient and the remainder, if necessary. Ask the students to find the quotient and the remainder of the following polynomials:

a.) (2y3 – 5y2 – 8y – 50) divided by (y – 5)

b.) (3y3 + 2y2 – 7) divided by (y – 3)

B. Introduce the lesson by reviewing how to find the zeroes of a polynomial function.

Recall the meaning of “zeroes of a polynomial function”. Ask the students for illustrations of quadratic and linear functions.

Ask them to extend the meaning of zeroes of polynomials of degree > 2 by illustrating this graphically.

Present the graphs of the following linear and quadratic functions, and ask the students to answer the guide questions that follow.

Sample Graphs of Linear Functions

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Graphs of Quadratic Functions

y=x2

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Guide Questions

For each function:

At what point/points does the graph cross the x-axis?

Notice that in each of the points of intersection with the y-axis the value of f(x) is zero. What is the value of x when f(x) = 0?

What are these x-values called?

1. Ask the students to find the value of x when f(x) = 0

a. Find the zero of f(x) = 3x – 5.

F(x) = 3x – 5 0 = 3x - 5 3x = 5

x = 5/3

b. Find the zeroes of f(x) = x2 – 5x + 4

F(x) = x2 – 5x + 4 x2 – 5x + 4 = 0 (x – 4)(x – 1) = 0

x = 4, x = 1

The zeroes of the function are 4 and 1

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2. Illustrate the procedure for finding the zeroes of a polynomial function by factoring. Let the students find the zeroes of the following functions:

a) P(x) = (x – 5)2 (x + 3) Zeroes are 5 and –3, 5 being a double root

b) P(x) = 3x3 + 9x2 – 30x Zeroes are 0, -5 and 2.

3. Recall the Factor Theorem, which states that:

“Given the polynomial f(x). If the remainder when f(x) is divided by (x-c) is zero, then (x-c) is a factor of f(x).

Main Activity

Part I

Let individual students try out some integers in the synthetic division for each polynomial and find the zeroes of the given P(x):

a.) P(x) = x3 + 9x2 + 23 x + 15

b.) P(x) = x4 + 5x3 + 5x2 – 5x – 6

Answers

a.) Zeroes are –1, -3, and –5

b.) Zeroes are 1, -1, -3, and –2.

Guide Questions

For P(x) = x3 + 9x2 + 23x + 15, what numbers may possibly be eliminated? (0 and positive values). Why?

How many negative integers should one try out as possible zeroes?

When should one stop dividing synthetically?

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Discussion Ideas

Based on the previous lesson, the students were able to find the factors of polynomials of higher degree. At this point, the students shall find the zeroes of such polynomials.

The polynomial f(x) = 4x3 – 4x2 – 8x is of the third degree; therefore, we expect the function to have three zeroes.

Look for a common monomial factor among the terms of the function 4x3 – 4x2 – 8x and factor this out: 4x (x2 - x – 2).

Note that x2 – x – 2 can be factored further as (x – 2)(x + 1).

Therefore, 4x3 – 4x2 – 8x = 4x (x – 2)(x + 1)

Equating all the factors to zero since f(x) = 0 we find that if:

4x = 0 x – 2 = 0 x + 1 = 0 Then:

x = 0 x = 2 x = -1

Note: This method is applicable if the given polynomial can easily be factored.

Part II. Present another example for the students.

f(x) = x4 – x3 – 7x2 + x + 6

The factors of 6 are: 2 and 3, -2 and -3, 6 and 1, -6 and -1?

Solution

Because f(x) is not readily factorable, we make use of a theorem called the RATIONAL ROOT THEOREM:

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Rational Root Theorem

Given a polynomial p(x0 = anxn+an-1xn-1 + … + a0. Then the rational

roots of the polynomial are of the form where bo and bn are

divisors of ao and an, respectively.

We apply this theorem to our examples:

fx) = x4 – x3 – 7x2 + x + 6 where, an = 1 and a0 = 6.

The divisors of ao = 6 are: 1, 2, 3, 6, -1, -2, -3, -6

The divisors of ao = 1 are: 1, -1

The only possible rational roots of f(x) are: 1, 2, 3, 6We choose which of these numbers to use in synthetic division:

Note 1. Clearly, 1 is a root, since f(1) = 1 – 1 – 7 + 1 + 6 = 0. Thus,

the remainder when f(x) is divided by (x-1) is zero. By the FACTOR THEOREM, (x – 1) is a factor of f(x).

2. Similarly, -1 is a root since f(-1) = 1 + 1 – 7 – 1 + 6 = 0

Because 1 is a root, we look for the quotient by Synthetic

Division

1 1 -1 -7 1 6

1 0 -7 -6

-1 1 0 -7 -6 0-1 1 6

1 -1 -6 0

Note: We can continue the synthetic division in this way until all the rational roots have been found.

f(x) = x4 – x3 – 7x2 + x + 6

= x3 – 7x - 6

= x2 – x – 6

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So far, we found two roots of f(x): 1 and –1.

Using the Factor Theorem, we know that:

f(x) = (x – 1)(x + 1) (x2 – x – 6)

= (x – 1)(x + 1) (x –3)(x + 2)

Thus, the roots of f(x) are 1, -1, 3 and –2.

Note: After the above example, the teacher may give another similar example for the students to work on by themselves.

Example

Find all the zeroes of f(x) = x4 + 2x3 – 3x2 – 8x – 4. Use all available techniques.

Solution

a) By the Rational Root Theorem, an = 1, ao = –4

possible rational roots 1, 2, 4.

b) By using the Remainder Theorem: we try to guess the roots

f(1) = 1 + 2 – 3 – 8 – 4 = 3 – 3 – 8 – 4 0 1 is not a root

Thus, x – 1 is NOT a factor of f(x))

f(-1) = 1 - 2 – 3 + 8 – 4 = 1 – 5 + 8 – 4 = 0

-1 is a root of f(x), and so (x + 1) is a factor of f(x)

c) Using Synthetic Division find the quotient:

-1 1 2 -3 -8 -4-1 -1 4 4

-1 1 1 -4 -4 0-1 0 4

f(x) = x4 + 2x3 – 3x2 – 8x - 4

(we use the remaining possible numbersto check if they are roots)

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1 0 -4 0

Note: It is possible that a number is a root more than once. In this case, -1 is of MULTIPLICITY 2 because it is a root 2 times

So far, we know by Synthetic Division and the Factor Theorem that:

f(x) = (x + 1)(x + 1)(x2 – 4) = (x + 1)(x + 1)(x + 2)(x – 2)

Hence, the roots of f(x) = x4 + 2x3 – 3x2 – 8x – 4 are –1, –2 and 2, where –1 is of multiplicity 2.

Part III. Divide the students into small groups. Ask them to find the zeroes of each of the given functions:

1. f( x ) = 2x3 – x2 – 13x – 6

2. f( x ) = 4x3 + 13x2 – 37x – 10

3. f( x ) = 6x3 + 25x2 + 2x – 8

4. f (x) = 6x3 + 19x2 + 11x - 6

Guide Questions

What are the ways of determining the zeroes of a function?

Without graphing, how are the zeroes of a function determined?

How many zeroes are there in a:

linear function?

quadratic function?

polynomial of degree n?

Closing Activity

Ask the students: What is meant by “zeroes of a function”?

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1. A zero of a polynomial function is the value of the variable x, which makes the polynomial equal to zero, or f(x) = 0.

2. The zeroes of P(x) are the roots of the equation P(x) = 0. A linear function has only one zero, while a quadratic function has at most two zeroes.

3. The zeroes of quadratic function can be found by factoring, completing the square, or by using the quadratic formula.

4. In general, a polynomial of degree n will have at most n real roots or zeroes.

ASSESSMENT

A. Factor the following polynomials into linear factors:

1. f(x) =x4 + 3x3 – 3x2 – 7x + 6

2. f(x) =x4 + x3 – 11x2 – 9x + 18

B. Find the zeroes of the following:

a. f(x) = x + 3

b. f(x) = x2 – x – 30

c. f(x) = x2 – 2

d. f(x) = x2 + 4x + 4

HOMEWORK

Graph the following functions:

1. f(x) = 2x – 7

2. f(x) = 3x + 8

3. f(x) = x2 – 8x – 48

4. f(x) = x2 + 9x - 22

REFERENCES

Foster, Gordon. Algebra 2 With Trigonometry–Applications and Connection

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78


Lesson 12GRAPHS OF POLYNOMIAL FUNCTIONS (Part

1)

TIMEOne session

SETTINGMath room

OBJECTIVESAt the end of the lesson, the students should be able to:

graph the polynomial functions of degree greater than two; and

locate the zeroes of a polynomial function by estimating its position on a graph

PREREQUISITEThe students must be familiar with the Factor Theorem, the Remainder Theorem, the Rational Root Theorem, and with synthetic and long division

RESOURCES

chart

PROCEDURE

Opening Activity

a) Check the students’ basic knowledge on evaluating a polynomial function. Ask them to find the value of P(x):

Given: P(x) = x3 – 3x2 – x + 10

Find: P(2), P(-2), P(3)

b) Introduce the new lesson by asking students to illustrate the graphs of the following:

a) F(x) = a?

b) F(x) = ax + b?

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c) F(x) = ax2 + bx + c?

d) F(x) = x3?

c) Allow the students to illustrate the different cases or possibilities for the first three. Let the students graph each of the functions.

Main Activity

Note! Prepare sample graphs for presentation to the students:

1. Point out the zeroes of the functions graphed on the board. Then, explain the zeroes of the polynomial functions.

2. Let the students observe the graphs of the polynomial function

f(x) = xn, when:

a) n is an integer greater than zero.

b) n is even

c) n is odd

3. Give some examples:

a. Consider the function f(x) = x4 – 5x2 + 4

Using the different techniques for finding the roots of a polynomial function, the roots of f(x) are found to be: –2, -1, 1 and 2.

The graph of f(x) looks like this:

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Notice that the graph of f(x) intersects the x-axis at the zeroes of the function (f(x)= x4 – 5x2 + 4) or where y is equal to zero. Because f(x) is of degree 4, it has, at most, 4 real roots.

b. On the other hand, the graph of f(x) = x4 – 2x3 – 2x – 1 looks like this:

How many zeroes does f(x) have? Try to verify your answer by using previous techniques learned.

4. Individual Activity

a) Ask the students to follow the steps in graphing the polynomial function: f(x) = (x – 2) (x + 1)(x – 1), as described in the textbook.

-1 1 -1

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Remind them that the zeroes of a polynomial function are the roots of the related polynomial equation.

b) Find the zeroes of the following polynomial functions, then graph.

P(x) = x3 – 9x + 1

P(x) = x3 – 4x - 2

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Discussion Ideas

1. Describe the graph of a polynomial function of degree greater than 2. Differentiate it from linear and quadratic functions.

2. What is meant by “the zeroes of a function”? How can you determine the zeroes from the graphs?

3. What does the graph of f(x) = xn look like when n is even? What if n is odd?

4. How would you describe the graph of f(x) = x5? What about f(x) = x6?

5. What is the basic step in graphing any function?

Closing Activity

Ask the students:

What can you say about the graphs of polynomial functions?

The graph of a polynomial function has the equation of the form f(x)=x n, where n is an integer greater than zero. It is continuous such that it has no breaks and has smooth rounded turns.

ASSESSMENT

Graph the function P(x) = x3 – 3x + 1, and estimate the real roots to the nearest one-half unit.

HOMEWORK

Use a graph to estimate the real roots to the nearest half unit of the equation P(x) = 0.

P(x) = x3 – x2 – 2x – 1

P(x) = x3 – x2 – 2x + 1

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REFERENCES

Foster and Gordon. Algebra 2 With Trigonometry–Applications and Connections.


Math IV: Advanced Algebra, Trigonometry and Statistics–2002 BEC


84



2)

TIME

One session

SETTING

This lesson can be done inside the math room. This lesson is supplementary to the previous lesson. The teacher should focus more on polynomial functions of degree >2 and not spend too much time recalling quadratic functions.

OBJECTIVES


sketch the graph of a given polynomial function by assigning points;

apply previously learned theorems and concepts in graphing polynomials of degree > 2.

PREREQUISITE

The students must have enough skills in applying the Factor Theorem, factoring techniques, synthetic division and depressed equations to find the zeroes of polynomial functions of degree greater than two.

RESOURCES

graphing boards graphing papers colored chalk chart


A. As a practice exercise, ask the students: Which of the following functions is linear? Which are quadratic?

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1. P(x) = 3x + 5

2. P(x) = x2 – 5x + 4

3. P(x) = x2 - 4

B. To review the graph of a polynomial function, show 5 samples of graphs of linear and quadratic equations. Let the students identify the kinds of graphs presented.

1) Present three linear functions, e.g.:

f(x) = x + 2

f(x) = -2x – 1

f(x) = x – 5

Then ask the students to draw the graphs of each of these and let them identify the following:

a. line

b. slope

c. x-intercept

d. y-intercept

Ask the students: How is the graph of a linear function drawn?

1) Present two quadratic functions, such as:

f(x) = x2 – 3x – 4

f(x) = -x2 + 2x + 24.

Give the students a few minutes to draw their graphs and let them identify the following:

a) parabola

b) vertex

c) axis of symmetry

d) direction of opening

e) x-intercepts

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Ask the students: How is the graph of a quadratic function drawn?

C Present a third-degree polynomial function in factored form, e.g., P(x) = (x – 2)(x +1)(x – 1). Prepare a Cartesian coordinate plane on the board and guide the students in sketching the graph of this function.

Main Activity

A. Compare the graphs of linear and quadratic functions:

From previous discussions, we learned of the graphs of two polynomial functions, the linear and the quadratic functions. The graph of a linear function f(x) = ax + b, where a and b are real numbers, is a straight line; the graph of a quadratic function, f(x) = ax2 + bx + c, where a, b, and c are real numbers and a is not zero, is a parabola

B. Allow each group to work on the following problems:

1. What do you think will happen if:

a) g(x) is decreased by 2 as in g(x)= x3 – 2?

b) h(x) is increased by 3 as in h(x) = x3 + 1

2. Sketch each of them and compare them with the graph of

f(x) = x3.

Discussion Ideas

a. Consider the following important points:

1) The zeros of a function are really the x-intercepts of the function. These can be plotted easily.

2) The values of the function in the following intervals will provide a picture of the behavior of the graph in these intervals.

x < -1

-1< x < 1

1 < x < 2

x > 2

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b. Have the students prepare a table of values where 2 to 3 values for x from each of the above intervals are used. These should be enough for the students to have an idea of the behavior of the graph in each interval. Let them show the table of values and the resulting graph:

Closing Activity

a. Ask the students: How do you graph a given polynomial function?

b. Summarize their responses:

To graph a polynomial function, first find the zeroes of the function. Then construct a table of values of the given variables and the corresponding value of the polynomial function. These values should represent points in the different intervals into which the zeroes of the function divide the x-axis. Plot these values on a Cartesian coordinate plane. If the points are rather far apart, assign fractional values to the variable to determine the shape of the curve more accurately.

ASSESSMENT1. Draw the graph of the polynomial function: f(x) = x3- 4x2 + x + 3

2. What are the roots of the equation?

3. Show the table of values.

4. What are the x-intercepts?

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HOMEWORK

1. Graph the function f(x) =

2. Graph the function f(x) = x3 + 2x2 – 4x – 1

3. Graph the function f(x) = (x – 1)4

REFERENCES



Math IV: Advanced Algebra, Trigonometry and Statistics – 2002 BEC


89




3)

TIME

One session

SETTING

Math room

OBJECTIVE


solve polynomial functions using the most logical procedure; and

graph polynomial functions of degree greater than two.

PREREQUISITE

Students must have enough skills in applying the Factor Theorem, factoring techniques, synthetic division and depressed equations to find the zeroes of polynomial functions of degree greater than two.

RESOURCES

graphing boards graphing paper colored chalk chart

PROCEDURE

Opening Activity

A. Have students review plotting of points. Ask them to identify the following features of the graph of a polynomial function:

a. vertexb. axis of symmetry

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c. x-intercepts

d. y-intercepts

B. The teacher may choose to solve items in the assignment.

Main Activity

Students have learned to identify the zeroes and turning points of a function from a prepared graph. With this, they can proceed to the next activity. Present the solution and graph of a polynomial function to the students for observation. Example: f(x) = x3 – 3x + 2

We can use the Rational Root Theorem to find out the possible roots of f(x). In f(x) = x3 –3x+ 2, the factors of 2 are 1, -1, 2, -2. We test each of these to determine the roots:

f(x) = x3 – 3x + 2f(1) = 13 – 3(1) + 2 = 1 – 3 + 2 = 0, then 1 is a zero

f(x) = x3 – 3x + 2f(-1)= (-1)3 – 3(-1) + 2 = -1 + 3 + 2 = 4, then –1 is not a zero

f(x) = x3 – 3x + 2f(2) = 23 – 3(2) + 2 = 8 – 6 + 2 = 4, then 2 is not a zero

f(x) = x3 – 3x + 2f(-2) = (-2)3 – 3(-2) + 2 = -8 + 6 + 2 = 0, then –2 is a zero

Ask the students to prepare a table of values using numbers representing the various intervals. Then, plot the graph of the function.

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Discussion Ideas

Ask the students to look at the graph and to answer the following questions:

1) What are the zeroes of the function?

2) What happens to the graph in each interval?

3) What are the factors of f(x)?

4) Which root has a multiplicity of 2?

Extension Ideas

Ask the students to look for a problem in science that make use of the zeroes of a polynomial function.

Closing Activity

Emphasize the key points of the lesson by asking:

1) How do you graph a given polynomial function?

2) How do you find the zeroes of a polynomial function?

ASSESSMENT

Solve and graph the function f(x) = x3 + 6x2 + x + 6

What are the zeroes of the function?

HOMEWORK

Solve and graph:

1. f(x) = x3 – 8

2. f(x) = x4 + 3x3 + 2x2 - 2

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REFERENCES

Foster and Gordon. Algebra 2 With Trigonometry—Applications and Connections,


Math IV: Advanced Algebra, Trigonometry and Statistics – 2002 BEC.

Mathematics IV – SEDP Series.

Yu- hico. Experiencing Mathematics 4.

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4)TIME

One session

SETTING

Math room

OBJECTIVES

At the end of this lesson, the students should be able to accurately draw the graph polynomial functions of degree greater than two.

PREREQUISITE

Students must have enough skills in applying the Factor Theorem, factoring techniques, synthetic division and depressed equations to find the zeroes of polynomial functions of degree greater than two.

RESOURCES

Manila paper graphing papers pencil ruler

PROCEDURE

Opening Activity

1. Check the students’ assignments.

2. Divide the class into 5. Let each group draw the graph of any given polynomial function on A3–sized bond paper. Let them emphasize the curve using a marker pen or crayon. Ask them to create an artwork using the curve as background.

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3. Ask each group to describe its graph. They may add something like:

In graphing polynomial functions, the zeroes of the polynomial function are the roots of the polynomial equation which corresponds to f(x). The real roots of any polynomial equation correspond to the point of intersection of its graph and the x-axis.

Main Activity

Group the students into 10. Give them 30 minutes to work on the given polynomials. Assign a leader in each group and a reporter to discuss the output of the group.

Tasks

1. Graph the polynomial function: f(x) = 4x3 + 16x2 + 9x – 9

2. Sketch the graph of the polynomial function: f(x) = x3 + 3x2 - x – 4

Closing Activity

a) Emphasize the key points of the lesson by asking:

1) How do you graph a given polynomial function?

2) How do you find the zeroes of a polynomial function?

b) Summarize their responses as follows:

To graph a polynomial function, first find the zeroes of the function. Then construct a table of values of the given variable and the polynomial function. These values should represent points in the different intervals into which the zeroes of the function divide the x-axis. Plot these values on a Cartesian coordinate plane. If the points are rather far apart, assign fractional values to the variable to determine the shape of the curve more accurately.

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HOMEWORK

1. Graph the function y = x2 + 3x2 –x –3 using (-3,-2,0,1,2,3) as the domain.

2. Approximate the real zeroes by graphing the polynomial function:

y = x3 – 4 x2 + x + 5

REFERENCES



Math IV: Advanced Algebra, Trigonometry and Statistics – 2002. BEC.

Mathematics IV – SEDP Series.

Yu-hico. Experiencing Mathematics.

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Unit Integration Plan Objectives

While studying polynomials, students can:

study drug absorption rates;

investigate how volcanic eruptions are predicted;

apply polynomials to situations involving population growth;

explore, analyze, and compare the benefits and liabilities of annuities;

analyze, solve and perform real-world problems.

Procedure

Ask each group to solve the following problems:

1. Given the prospective earnings and the prevailing interest rate, determine the amount that will be available for graduate school expenses if a college student saves income from summer employment following high school, until he/she enters graduate school after four years of college.

2. Given the sum of the length, width, and depth of carry-on luggage, determine the real domain for these variables as well as the maximum volume of a carry-on item if the length is to be 10 centimeters more than the depth.

3. Plot the given data on the number of minutes needed to have Caucasian skin tanned at different times of the day in Boracay. Use a graphing calculator to determine a polynomial function to graph the data. Use the function to determine the time of day when skin will tan in 30 minutes. Explain why the time would be the same or different in different parts of the Philippines.

4. Given three patterns for constructing closed rectangular containers

from pieces of cardboard, determine which of the three patterns produces a container with maximum volume and minimum waste.

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Using the computed values, students can be asked to apply their mathematical skills in making bags, ornamental boxes and useful containers which they can sell. They can use any available resources to produce these products and their creativity in making the best appropriate design. Accuracy and quality of work are the main criteria for evaluation.

The products produced may be accompanied by a marketing plan, so students can sell it later and earn some profit.

5. Students may interview professionals in various fields of study (e.g., economics, engineering, architecture, social sciences, etc. to find out how they make use of polynomials in their work.

As a final project, they may work with these professionals with the supervision of the teacher or some other adult or student’s guardian. In this way, students may have more appreciation for mathematics, and develop work ethics at the same time.

Assessment

In a scale of 6, where 6 is the highest score and 1 is the lowest score, rate the overall performance of the students in the culminating activity.

6 Exemplary response

gives a complete response with a clear, coherent, unambiguous and elegant explanation;

includes a clear and simplified diagram; communicates effectively to the identified audience; shows understanding of the open-ended problem's

mathematical ideas and processes; identifies all the important elements of the problem; may include examples and counterexamples; presents strong supporting arguments.

5 Competent response

gives a fairly complete response with reasonably clear explanations;

may include an appropriate diagram;

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communicates effectively to the identified audience;

shows understanding of the problem's mathematical ideas and

processes; identifies the most important elements of the

problem; presents solid supporting arguments.

4 Minor Flaws But Satisfactory completes the problem satisfactorily, but the

explanation may be muddled;

argumentation may be incomplete; diagram may be inappropriate or unclear;

understands the underlying mathematical ideas;

uses mathematical ideas effectively.

3 Serious Flaws But Nearly Satisfactory begins the problem appropriately but may fail to

complete or may omit significant parts of the problem;

may fail to show full understanding of mathematical ideas and processes;

may make major computational errors;

may misuse or fail to use mathematical terms;

response may reflect an inappropriate strategy for solving the problem.

2 Begins, But Fails to Complete Problem explanation is not understandable;

diagram may be unclear;

shows no understanding of the problem situation;

may make major computational errors.

1 Unable to Begin Effectively words do not reflect the problem;

drawings misrepresent the problem situation;

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copies parts of the problem but without attempting a solution;

fails to indicate which information is appropriate to then problem.

Source http://intranet.cps.k12.il.us/Assessments/Ideas_and_Rubrics/Rubric_Bank/MathRubrics.pdf

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