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Honors Algebra 2 Unit 4: Chapter 6SPRING 2014: POLYNOMIAL FUNCTIONSNC OBJECTIVES: 1.02 Define and compute with complex numbers.1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems. 2.04 Create and use best-fit models to solve problems.2.06 Use cubic equations to model and solve problems. a) solve using graphs. b) Interpret constants and coefficients. Day Date Lesson Assignment

1 Mon.March 17

“What do you know about Polynomials?”Section 6.1: Classify polynomials, end behavior,

and difference to find the degreePacket p. 1

2 Tues.March 18 Section 6.2: Polynomials and Linear Factors Packet p. 2(even)

3 Wed.March 19

Solve Polynomial equations by factoring & graphing

Factor CubesPacket p. 3

4 Thurs.March 20

Review of 6.1 & 6.2(packet p. 4 & 5)Section 6.3: Dividing Polynomials(long division)

Synthetic DivisionPacket p. 6

5 Fri.March 21

Section 6.5: Finding factorsReview for mid-chapter test

Packet p. 7 #1-7Review Sheet

STUDY for TEST!

6 Mon.March 24

Mid-chapter Test(1/2 of a test grade)Review for opportunity for mastery Mastery Opportunity Review

Sheet7 Tues.

March 25Section 6.5: Finding all zeros &

Writing Equations Given the ZerosPacket p. 7 Parts IV and V

Packet p. 8

8 Wed.March 26

Polynomial Models in the real world Transforming Polynomials

Opportunity for Mastery TestPacket p. 9(Top Half)

9 Thurs.March 27

Expanding binomials using Pascal’s Triangle Review for Unit Test: Polynomial Functions

Packet p. 9(bottom half)

Test Review: PacketPages 10 & 11

10 Fri.March 28

Unit Test 4: First test of the new QuarterCounts ½ of a test grade

Packet Pgs. 12 & 13Print Unit 5 Notes &

Packet!!

Write each polynomial in standard form. Then classify it by degree and by number of terms.

1) -3 + 3x – 3x 2) x2 + 3x – 4x3 3) a3 (a2 + a + 1)

4) p (p – 5) + 6 5) (3c2)2 6)

Determine the end behavior of the graph of each polynomial function.

7) 8) 9)

Determine the degree of the polynomial function with the given data.

10) 11)

Describe the shape of the graph of each cubic function by determining the end behavior and number of turning points.

12) 13)

Determine the sign of the leading coefficient and the degree of the polynomial function for each graph.

14) 15) 16)

Homework Day 2: Polynomials and Linear Factors

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x y-2 -16-1 10 41 52 16

x y-2 52-1 60 21 42 48

Factor the sum or difference of cubes.

Factor the expression on the left side of each equation. Then solve the equation.

13. 14. 15.

16. 17.

Solve each equation by graphing. Where necessary, round to the nearest hundredth.

18. 19. 20.

Solve each equation.

21. 22. 23.

Graphing Polynomial Equations

2

Determine the degree of each polynomial equation given below. Then list the roots and their multiplicity. Lastly, match each polynomial equation to its correct graph shown below. The one is done for you.

Equation Graph Degree Roots/Multiplicity

D 6Root 0 -2 3Mult

.1 2 3

Root Mult.RootMult.RootMult.RootMult.RootMult.RootMult.RootMult.RootMult.RootMult.

A B

C

3

Graphing Polynomial Functions(continued)

ED F

G H

I

J

4

Part 1: Divide using long division 1. (x2 – x – 56) (x + 7) 2. (6x3 – 11x2 – 47x - 20) (2x + 1)

3. (2x3 – 2x – 3) (x – 1) 4. (2d3 – 5d2 + 18) (2d + 3)

Part 2: Divide using synthetic division

5. (x2 – 5x – 12) (x + 3) 6. (2x3 + 3x2 – 8x + 3) (x + 3)

7. (x4 + x3 – 1) (x – 2) 8. (x3 – 9x2 + 27x - 28) (x - 3)

Part 3: Find each remainder. Is the divisor a factor of the polynomial?

9. (10x3 – x2 + 8x + 29) (5x + 2) 10. 2x4 + 14x3 – 2x2 – 14x) (x + 7)

Part 4: Use synthetic division to find the value of the function.

Part 5: Application12. A box is to be mailed. The volume in cubic inches of the box can be expressed as the product of its three dimensions: . The length is (x - 8). Find linear expressions for the other dimensions. Assume that the width is greater than the height.

13. An open box is made from an 8-by-10-inch rectangular piece of cardboard by cutting squares from each corner and folding up the sides. If x represents the side length of the squares, write a function giving the Volume V(x) of the box in terms of x?

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Honors Algebra 2 ~ Finding All Zeros HomeworkI. Use synthetic substitution to find f(4) for each function.

1. f(x) = x3 + 2x2 – 3x + 1 2. f(x) = 5x4 – 6x2 + 2

II. Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials.

3. x3 – x2 – 5x – 3; x + 1 4. x3 + x2 – 16x – 16; x – 4

5. 2x3 + 17x2 + 23x – 42; 2x + 7

III. Use the graph of each polynomial function to determine at least one binomial factor of the polynomial. Then find all of the factors.

6. f(x) = x5 + x4 –3x3 – 3x2 – 4x – 4 7. f(x) = x4 + 7x3 + 15x2 + 13x + 4

IV. Find all of the zeros of each function

8. f(x) = 6x3 + 5x2 – 9x + 2 9. f(x) = 6x4 + 22x3 + 11x2 – 38x – 40

10. p(x) = x3 + 2x2 – 3x + 20 11. f(x) = x3 – 4x2 + 6x – 4

V. Think about a Plan: A polynomial function, , is used to model a new roller coaster section. The loading zone will be placed at one of the zeros. The function has a zero at 5. What are the possible locations for the loading zone? What methods can you use to solve this problem? SOLVE.

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Honors Algebra 2 ~ Roots and Zeros HomeworkI. Given a function and one of its zeros, find all of the zeros of the function

1. f(x) = x3 – 4x2 + 6x – 4; 2 2. h(x) = 4x4 + 17x2 + 4; 2i

3. f(x) = x3 – 7x2 + 16x – 10; 3 – i 4. r(x) = x4 – 6x3 + 12x2 + 6x – 13; 3 + 2i

II. Write the polynomial function of least degree with integral coefficients that has the given zeros.

5. –2, 1, 3

6. 4i, 3, -3

7. 4, 2 + i

8. 5 + 2i, -2

9. 3, -4i

10. -4, 2 – 3i

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1. The following table shows the percent of on-time flights for selected years. Find the best fit model for the following data. According to the model, what is the percentage going to be for the year 2012?

Year 1998 2000 2002 2004 2006On-Time Flight Percentages 77.20 72.59 82.14 78.08 75.45

2. The table at the right shows the number of students enrolled in a high school personal finance course. Find the best fit model for the following data. According to the model, predict the enrollment for 2012.

Year Number of Students Enrolled2000 502004 652008 942010 110

Homework: Day 9Use Pascal’s Triangle to expand each binomial.(#7, 9, 10)

Expand Each Binomial. #25-35 multiples of 5.

Honors Algebra 2 ~ Review for Unit 4 Test1. Find the remainder for (x3 + 2x2 – 4x – 5) (x – 2)

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2. Find all of the zeros for f(x) = x3 – 19x – 30

3. Find the remaining factors of x3 – x2 – 5x – 3 if x + 1 is a factor

4. Find the remaining factors of x3 – 4x2 + 12x – 27 if x – 3 is a factor

5. What are all of the zeros of f(x) = x3 + 7x2 + 25x + 175 if 5i is a zero?

6. What is the polynomial function of least degree with integral coefficients that has 2 and 4i as its roots?

7. What is the polynomial function of least degree with integral coefficients that has 9 and (1 + 2i) as its roots?

8. What is the polynomial function of least degree with integral coefficients that has (1 – 2i) and –3 as its roots?

9. Use synthetic substitution to find f(-3) for f(x) = 4x3 – 2x2 + x – 5

10. Use synthetic substitution to find f(5) for f(x) = 5x4 – 2x2 + 1

11. Solve: x4 – 12x2 – 45 = 0

12. Find all of the zeros for f(x) = 6x3 + 5x2 – 9x + 2

13. Find all of the zeros for p(x) = 6x4 + 22x3 + 11x2 – 38x – 40

14. Write an equation for each graph below in factored form and state the degree of each polynomial.

a. b.

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15. Use your calculator to find a cubic and a quartic function to model the data below. Which is the better fit? Using the better model, estimate the sales in 1994.

Years since 1980

3 5 7 8 10 12 15

Answering Machines sold (in millions)

2 4.2 8.8 11.1 13.8 16 17.7

16. Expand (4x – 3y)5 using Pascal’s Triangle.

17. Expand (2x + 5y)4 using Pascal’s Triangle.

Review Sheet Answers:1. remainder is 32. x = -3, x = -2, x = 53.4.5.6.7.8.9.10.11.

12.

13. 14. curves through –2(mult. 3),up, up, passes through at 2(mult 1) a) degree 4

b) bounces at –1(mult 2) passes through 4(mult 1) down, up , degree 3

15. cubic: quartic: quartic is the best fit16.84 million in 1994

16. 17.

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Unit 4 Cumulative Review Questions

___________1. Solve for X. [0 0 21 3 −21 −2 1 ]X=[ 6

−118 ]

A. [ 1−23 ]

B. [16−4336 ]

C. [9 . 82.4

3 ]D.

[1623−8]

___________2. Write the quadratic equation in standard form of a parabola containing the following points: (-3,-4), (-1,0) and (9.-10)

__________3. Simplify 1+2i2−3 i .

A. 8+i7 B.

−4+7 i13 C.

8+7 i7 D. −4+7 i

__________4. Which equation for the parabola has vertex (-2,3) and passes through (-1,5)?

A. y = x2 + 4x + 7 B. y = x2 -4x + 7 C. y = 2x2 -8x + 11 D. y = 2x2 + 8x + 11

__________5. A ball is thrown upward, its height in feet is given by: h(t) = -16t2+64t + 3, where t is time In seconds. What is the maximum height that the ball reaches?

A. 3 feet B. 51 feet C. 63 feet D. 67 feet__________6. What is the discriminant of 3x2 – 14x + 9 ?

A. 2√22 B. -94 C. 88 D. 2

__________7. Solve x2 + 8x = 6.

A. {−4±√11 } B. { 4±√10 } C. {−8±√22 } D. {−4±√22 }____________8. Factor completely: y

3−64

A. ( y−4 )3 B. ( y−4 ) ( y2+4 y+16 ) C. ( y−4 ) ( y+4 )2 D. ( y−8 ) ( y2+16 y+64 )___________9. Solve x2 – x – 12 ¿ 0.

A. [-4,3] B. [-3,4] C. (−∞ ,−3 ]∪[ 4 ,∞) D. (−∞ ,4¿∪[ 3 ,∞)¿_____________10. One of the factors of 21x2 – 2x – 3 is:

A. 7x – 3 B. 7x + 1 C. 3x – 1 D. 7x + 3

_____________11. Which of the following is not factorable.

A. a2 – b2 B. a2 + b2 C. a3 – b3 D. a3 + b3

____________12. Determine the degree of the following polynomial: 9x4 + 4x3y2 – 7x2y2 + 16xy5

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A. 4 B. 9 C. 6 D. 19____________13. Simplify (x + 3)(x2 + 5x – 4) .

A. x3 + 8x2 + 11x -12 B. x3 + 5x -12 C. x3 + 8x2 – 11x -12 D. x3 + 8x2 + 4x – 12

____________14. Divide using synthetic division: (2x4 + 6x3 + 5x – 6) ¿ (x + 2).

A. 2 x3+2 x2−4 x+13−32

x+2 B. 2 x3+2 x2+ x−8 C. 2 x3+2 x2+ x−32

x+2 D. 2 x3+2 x2+ x− 8

x+2

____________15. Divide using long division: (4 x3−2 x2+8 x−8 )÷(2x+1) .

A. 2 x2+4−12

2x+1 B. 2 x2+4− 9

2x+1 C. 2 x2−2 x+5−13

2x+1 D. x2−4 x+6−14

2 x+1

___________16. What are the x-intercepts of the graph of the function y = -3(x – 2)(x + 7) ?

A. 2 and -7 B. -2 and 7 C. -6 and 21 D. 6 and -21

___________17. What is the correct factorization of 4x2 + 14x – 8?

A. (2x – 1)(2x + 4) B. 2(2x + 1)(x – 4) C. 2(x + 4)(2x - 1) D. 2(2x + 4)(x – 1)

__________18. Which value is NOT a solution to the equation:

A. -3 B. 3 C. -3i D.

__________19. Which translation takes to ?A. 2 units right, 3 units down B. 2 units right, 3 units up

C. 2 units left, 3 units up D. 2 units left, 3 units down

__________20. Which way does this parabola open y = -4 ( x + 2 ) - 3 opens?A. Up B. Down C. Left D. Right

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