math analysis honors unit 4 polynomial functions

Math Analysis Honors – Unit 4 – Polynomial Functions

Tuesday 9/18 Today’s Topic: End Behavior of Polynomial Functions

In-class examples: Complete the following statements for each function:

( )

( )

As ____.

As ____.

,

,

x x

x x

f

f

→ →

→− →

1. (a) ( ) 32f x x= (b) ( ) 32f x x= − (c) ( ) 5f x x= (d) ( ) 0.5f x x= −

2. (a) ( ) 43f x x= − (b) ( ) 40.6f x x= (c) ( ) 62f x x= (d) ( ) 20.5f x x= −

Describe the patterns you observe. In particular, how do the values of the coefficient an and the degree n affect the end

behavior of ( ) n

nf x a x=

Homework: Worksheet 16

Wednesday 9/19 Today’s Topic: Identify the Zeros of a Polynomial Function and their Multiplicity

In-class examples:

The Factor Theorem: A polynomial function ( )f x has a factor x − k if and only if ( ) 0f k = , where k is an x-

intercept of the graph of ( )f x .

Ex. 1 Find a polynomial of degree 3 whose zeroes are −3, 2, and 5 . Draw a possible graph of this function.

Ex. 2 Is 1x − a factor of ( ) 3 2 1f x x x x= − + − ?

Ex. 3 For the polynomial ( ) ( )( )4

2 15 2 3

2f x x x x

= − + −

, state the zeroes and their multiplicities. Draw a

possible graph of ( )f x .

Ex. 4 For the polynomial ( ) ( )( ) ( )2 3

2 1 2 3f x x x x= − − + − ,

a) State the degree of ( )f x .

b) State each zero, determine whether it is of odd or even multiplicity.

c) Draw a possible graph of ( )f x .

Homework: Worksheet 17 and MathXL

Friday 9/21 Today’s Topic: Finding the Zeros of a Polynomial Function by Factoring

In-class examples:

Ex. 1 Find the zeros (roots) of ( ) 3 2 6f x x x x= − −

Ex. 2 Find the zeros (roots) of ( ) 3 36f x x x= −

Ex. 3 Find the zeros (roots) of ( ) 3 23 2f x x x x= − −

Ex. 4 Find the zeros (roots) of ( ) 3 23 4 12f x x x x= − − +


Monday 9/24 Today’s Topic: Graphing Polynomials using x-intercepts (with multiplicity), y-intercept and end

behavior.

In-class examples:

Ex. 1 Graph ( ) ( ) ( )3 2

2 1f x x x= + − by finding the x-intercepts, y-intercept and end behavior.

Ex. 2 Graph ( ) 3 23 2f x x x x= − − by finding the x-intercepts, y-intercept and end behavior.

Ex. 3 Graph ( ) 4 25 4f x x x= − +


Tuesday 9/25 Today’s Topic: Quadratics and Polynomials Review – Test Tomorrow

In-class examples:

Review the following topics: Graphing Quadratics; Solve Applied Quadratic Problems; End Behavior of a

Polynomial Function; Zeroes of a Polynomial; Graphing Polynomials


Wednesday 9/26 Today’s Topic: Quadratics and Polynomials Mid-Unit Quiz

In-class examples:

Good Luck on Today’s Quiz

Homework: None

Thursday 9/27 Today’s Topic: Rational Zeros Theorem: Using the constant and the leading coefficient, we

can develop a list of all potential rational (fractional) zeros

In-class examples:

Ex.1 List the possible rational zeros for the function, ( ) 3 23 4 5 2f x x x x= + − − . Find all rational zeros.

Ex. 2 List the possible rational zeros for the function, ( ) 3 22 9 9f x x x x= − − + . Find all rational zeros.

Ex. 3 Find all zeros of ( ) 4 3 23 2f x x x x x= + − − + .


Friday 9/28 Today’s Topic: Complex Numbers

In-class examples:

Recall: The imaginary number: 1i = − ; 2 1i = − ; 3i i= − ; 4 1i =

Example: 39 ?i =

Adding and Subtracting Complex Numbers

1) ( ) ( )7 3 4 5i i− + + 2) ( ) ( )2 8 3i i− − + 3) ( )8 4 3i i− −

Multiplying Complex Numbers

4) ( )( )2 3 5i i+ − 5) 3− 5i( )4i

Complex Conjugates

6) ( )( )4 5 4 5i i+ − 7) ( )( )1 7 1 7i i− +

Division

8) 5 + i

2 − 3i

9) Solve x2 + x +1= 0

Homework: Worksheet 22 (1-7)

Monday 10/1 Today’s Topic: Complex Zeros: If a bi+ is a zero of ( )f x , then a bi− is also a zero.

In-class examples: Given the zeros, write the polynomial function with least degree.

1) Zeros: 2 5 , 2 5x i x i= − = +

2) Zeros: 3, 1 , 1x x i x i= = + = −

Homework: Worksheet 22 (8-14) and MathXL

Tuesday – 10/2 – ACT - Seniors

Wednesday 10/3 Today’s Topic: Fundamental Theorem of Algebra: A polynomial function of degree n has n

complex zeros (real and nonreal). Some of these zeros may be repeated.

Complex Conjugate Zeros

In-class examples:

Ex. 1 One zero of ( ) 4 3 25 3 43 60f x x x x x= − − + − is 2 − i . Find the other zeros (real and nonreal)

Ex. 2 Write a polynomial function of minimum degree in standard form with real coefficients whose zeros

include x = 1 and x = 3+ i .

Ex. 3 Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and

their multiplicities are x = 1 (multiplicity 2) and x = 3+ i (multiplicity 1)

Homework: MathXL

Thursday 10/4 Today’s Topic: Finding Complex Zeros

In-class examples:

Ex. 1 The complex number 1 2x i= − is a zero of ( ) 4 24 17 14 65f x x x x= + + + . Find the remaining zeroes of

( )f x , and write it in factored form.

Ex. 2 Find all of the zeros and write a linear factorization of ( )f x if 1x i= + is a zero of

( ) 4 3 22 6 6f x x x x x= − − + − .

Ex. 3 Find all zeros of ( ) 5 4 3 23 5 5 6 8f x x x x x x= − − + − + .


Friday 10/5 Today’s Topic: Finding Complex Zeros

In-class examples:

Ex. 1 Write ( ) 5 4 3 23 2 6 4 24 16f x x x x x x= − + − − + as a product of linear and irreducible quadratic factors, each

with real coefficients.

Ex. 2 Write ( ) 3 2 2f x x x x= − − − as a product of linear and irreducible quadratic factors with real coefficients.

Ex. 3 Write a polynomial function of minimum degree in standard from with real coefficients whose zeroes

include: −1, 2, and 1+ i .


Monday 10/8 Today’s Topic: Polynomial Functions Review

In-class examples:

Homework: MathXL – Unit 4 Review

Tuesday 10/9 Today’s Topic: Polynomial Functions Test

In-class examples:

Good Luck on Today’s Test

Homework: None – Good Luck on the SAT or PSAT, or whatever test you are taking tomorrow!

Math Analysis Honors – Worksheet 16 End Behavior of a Polynomial Function

End Behavior and Zeroes of Polynomials.

Describe the end behavior of the graph of the polynomial function.

1 ( ) 42 3 1f x x x= − +

2 ( ) 27

5 32

g x x x= − −

3 ( ) 31

53

f x x x= +

4 ( ) 7 5 3 23 1 5 3

4 2 4 2f x x x x x= − + +

5 ( ) 61f x x= −

Sketch the graph of a polynomial function that satisfies the given conditions

6 Third-degree polynomial with two real zeros and a negative leading coefficient.

7 Fourth-degree polynomial with two real zeros and a positive leading coefficient.

Find all the real zeros of the polynomial function using a graphing calculator.

8 ( ) 2 2f x x x= + −

9 ( ) 4 3 220f x x x x= − −

10 ( ) 4 22 2 40f x x x= − −

11 ( ) 3 24 25 100f x x x x= − − +

Math Analysis Honors– Worksheet 17 Zeros of Polynomials

Find a polynomial function of the least degree possible that has the given zeros. Draw a possible sketch of the

polynomial.

1 0, 4

2 -7, 2

3 -2, -1, 0, 1, 2

4 2, 5,4 5−

Find a polynomial function with the given zeros, multiplicity, and degree. You may leave your answers in factored form.

(There are many correct answers). Draw a possible graph of ( )f x .

5 Zero: -2, multiplicity: 2

Zero: -1, multiplicity: 1

Degree: 3

Positive leading coefficient

6 Zero: -5, multiplicity: 3

Zero: 0, multiplicity, 2

Degree: 5

Negative leading coefficient

7 Find a polynomial function in standard form with a degree of 4 and zeros at 1, and 2.

8 Find a polynomial function f of degree 3 such that ( )10 17f = and the zeros of ( )f x are 0, 5, and 8.

Use the factor theorem to determine whether the first polynomial, ( )q x is a factor of the second polynomial, ( )f x If so,

write ( )f x in factored form.

9 ( ) ( ) 3 23; 15q x x f x x x x= − = − − −

10 ( ) ( ) 3 22; 4 9 3 2q x x f x x x x= + = + − −

11 Given that 3x = − is a zero of ( ) 3 25 7 49 51f x x x x= − − + , find the other two zeros.

Answers

1. ( ) 2 4f x x x= − 2. ( ) 2 5 14f x x x= + − 3. ( ) 5 35 4f x x x x= − +

5. ( ) 3 25 8 4f x x x x= + + + 6.

( ) 5 4 3 215 75 125f x x x x x= − − − −

7. Answers will vary 8. ( ) 3 217 221 34

100 100 5f x x x x= − +

9. 5k =

Math Analysis Honors – Worksheet 18 Real Zeros of Polynomial Functions

Find the real zeros of the function. Write the function in factored form. You may use a calculator to find

enough zeros to reduce your function to a quadratic equation using synthetic substitution.

1 ( ) 3 22 13 24 9f x x x x= − + −

2 ( ) 3 28 17 6f x x x x= − + −

3 ( ) 3 24 4f t t t t= − +

4 ( ) 4 3 220f x x x x= − −

5 ( ) 3 24 4 7 2f x x x x= + − +

6 ( ) 4 3 26 4 8f x x x x x= − − + +

7 ( ) 3 23 3 9f x x x x= + − −

8 ( ) 3 2 8 6f x x x x= + − −

9 Find k such that ( ) 3 2 2f x x kx kx= − + + has a factor of x − 2

10 What is the remainder when ( ) 20 102 8 2f x x x x= − + − is divided by x − 1? (Hint: Do not use synthetic or

long division)

11 One solution of the equation 3 28 16 3 0x x x− + − = is 3 . Find the sum of the remaining solutions.

12 List the zeros of the polynomial, ( ) ( ) ( ) ( )2 3 2

2 1 5f x x x x x= − + − and their multiplicities. Draw a possible

sketch of ( )f x .

Math Analysis Honors – Worksheet 19 Graphing Polynomial Functions

Math Analysis – Worksheet 20 Quadratics and Polynomials Review

1 Sketch the graph of the quadratic function f x( )= 3x2−12x +11 . Identify the vertex, x-intercept(s) and y-intercept

without using a calculator

2 Write the vertex form of the quadratic function that has a vertex at ( )1, 4− and passes through the point ( )2, 3− .

3 Use a graphing calculator to graph the quadratic function f x( )= 30 + 23x + 3x2

. Find the x-intercepts of the graph.

4 Use a graphing calculator to graph the quadratic function ( ) 28 11g x x x= + + . Find the x-intercepts of the graph.

5 Find two numbers whose sum is 36 and whose product is as large as possible.

6 A husband and wife have enough wire to construct 160 ft. of fence. They wish to use it to form three sides of a

rectangular garden, one side of which is along a building. Find the dimensions that will yield the largest area.

7 A ball is thrown upward from the top of a 64-foot tower with an initial velocity of 96 ft/sec. The height of the ball is

determined by h t( )= −16t 2 + 96t + 64 .

a) How high is the ball after 5 seconds?

b) When does the ball reach its maximum height?

c) How high will the ball go?

d) When will the ball reach the ground?

8 Write the polynomial function of least degree with zeroes at 2, 5 and - 5 .

Find the following for each of the functions in #9 and #10.

- End behavior

- x-intercepts (with multiplicities)

- y-intercept

- Extrema

- Intervals of Increase/Decrease

9 ( ) 3 25 26 5f x x x x= + +

10 ( ) 4 213 36f x x x= − +

11 Given that x = 2 is a zero of ( ) 3 22 5 6f x x x x= + − − , find the remaining zeros.

12 Determine if x + 1 is a factor of ( ) 3 28 11 20f x x x x= + + − . If so, factor ( )f x completely.

Math Analysis Honors – Worksheet 21 Rational Zeros Theorem

Find all the real zeros of the function, finding exact values whenever possible. Write the function in factored form.

1 ( ) 3 26 7 4f x x x x= − + +

2 f x( )= x4 − 3x3 − 6x2 + 6x + 8

3 f x( )= 2x3 − 3x2 − 4x + 6

4 f x( )= 2x4 − 7x3 − 2x2 − 7x − 4

5 f x( )= x3 + x2 − 8x − 6

Math Analysis Honors – Worksheet 22 Complex Numbers

Adding and Subtracting Complex Numbers

1) ( ) ( )7 3 4 5i i− + + 2) 2 − i( )− 8 + 3i( ) 3) 8i − 4 − 3i( )

Multiplying Complex Numbers

4) ( )( )2 3 5i i+ − 5) 3− 5i( )4i

Complex Conjugates

6) 4 + 5i( ) 4 − 5i( ) 7) ( )( )3 2 3 2i i− +

8 Write the polynomial in standard form: ( ) ( ) ( )2 2f x x i x i= − + − −

9 Write the polynomial in standard form: ( ) ( )( )( )( )3 3f x x x x i x i= − + − +

10 Write the polynomial in standard form: f x( )= x − 2i( ) x + 2i( )

11 Solve x2 + 2x + 5 = 0

12 Use synthetic division to verify that x = 2i and x = −i are zeros of f x( )= x2 − ix + 2 .

13 One zero of ( ) 3 23 25 75g x x x x= + + + is 5x i= − . Find the other zeros.

14 Write a 4th degree polynomial with zeros: 3 2 ; 4 (multiplicity of 2)i+ .

Math Analysis Honors – Worksheet 23 Fundamental Theorem of Algebra & Complex Conjugate Zeros

Write a polynomial function of minimum degree in standard form with real coefficients whose zeros include those listed.

1 1− 2i and 1+ 2i

2 −1, 2, and 3i

3 5 and 3+ 2i

Using the given zero, find all of the zeros and write a linear factorization of f x( ).

4 1+ i is a zero of f x( )= x4 − 2x3 − x2 + 6x − 6

5 1+ 3i is a zero f x( )= x4 − 2x3 + 5x2 +10x − 50

Find all of the zeros and write a linear factorization of the function.

6 f x( )= x3 + 4x − 5

7 f x( )= x4 + x3 + 5x2 − x − 6

8 f x( )= 3x4 + 8x3 + 6x2 + 3x − 2

Answers:

1) f x( )= x2 − 2x + 5 2) f x( )= x4 − x3 + 7x2 − 9x −18

3) ( ) 3 211 43 65f x x x x= − + − 4) ( ) ( ) ( ) ( )( )1 1 3 3f x x i x i x x= − + − − − +

5) ( ) ( ) ( ) ( )( )1 3 1 3 5 5f x x i x i x x= − + − − − + 6) f x( )= x −1( ) x +

1

2−

19

2i

x +

1

2+

19

2i

7) ( ) ( )( )1 23 1 23

1 12 2 2 2

f x x x x i x i

= − + + − + +

8) f x( )= x + 2( ) x −1

3

x +

1

2−

3

2i

x +

1

2+

3

2i

Math Analysis – Worksheet 24 Solving Polynomial Functions

List the zeros of the polynomial and state the multiplicity of each zero.

1. ( ) 54 4

5f x x x

= +

2. ( ) ( )

14 13152 11g x x x x= − −

Find all (including imaginary) zeros of f.

3. ( ) 23 2 7f x x x= + + 4. ( ) 3 125f x x= + 5. ( ) 4 22 7 4f x x x= − −

Find a polynomial function ( )f x that satisfies the given conditions. Write the function in standard form.

(Note: If 3 4i− is a zero of ( )f x , then its conjugate, 3 4i+ is also a zero)

6. zeros include 2 and 2i i+ − 7. degree 3; ( )1

zeros include 1, ,2; 0 22

f− =

8. zeros include 3, 3i 9. degree 2; zeros include 3 i+

One zero of the polynomial is given; find all zeros.

10. ( ) 3 22 2 3; zero 3f x x x x= − − − 11. ( ) 4 3 25 6; zero f x x x x x i= − − − −

12. ( ) 4 3 24 6 4 5; zero 2f x x x x x i= − + − + −

13. Find all zeros of ( ) 4 3 25 4 2 8f x x x x x= − + + − ; then write ( )f x in its factored form, with linear and irreducible

quadratic factors.

14. Find all zeros of ( ) 3 2 8 12f x x x x= − − + ; then write ( )f x in its factored form, with linear and irreducible quadratic

factors.

math analysis honors unit 4 polynomial functions

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