Name________________________________

Honors Algebra 2 Unit 4: Chapter 6 SPRING 2014: POLYNOMIAL FUNCTIONS NC 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 degree

Packet 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 Cubes Packet p. 3

4 Thurs.

March 20

Review of 6.1 & 6.2(packet p. 4 & 5)

Section 6.3: Dividing Polynomials(long division)

Synthetic Division

Packet p. 6

5 Fri.

March 21

Section 6.5: Finding factors

Review for mid-chapter test

Packet p. 7 #1-7

Review 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

Sheet

7 Tues.

March 25

Section 6.5: Finding all zeros &

Writing Equations Given the Zeros Packet p. 7 Parts IV and V

Packet p. 8

8 Wed.

March 26

Polynomial Models in the real world

Transforming Polynomials

Opportunity for Mastery Test

Packet 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: Packet

Pages 10 & 11

10 Fri.

March 28

Unit Test 4: First test of the new Quarter

Counts ½ of a test grade

Packet Pgs. 12 & 13

Print Unit 5 Notes & Packet!!

1

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) 42 4 5

4

x x

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

7) 4 3 23 6 12y x x x 8)

4 712 3 1y x x x 9) 2 2y x x

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) 32 3 1y x x 13)

3 25 6y x x

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

graph.

14) 15) 16)

x y

-2 -16

-1 1

0 4

1 5

2 16

x y

-2 52

-1 6

0 2

1 4

2 48

2

Homework Day 2: Polynomials and Linear Factors

3

Factor the sum or difference of cubes.

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

13. 327 1 0x 14. 38 27 0x 15. 4 28 16 0x x

16. 32 54 0x 17. 4 210 9 0x x

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

18. 4 22 9 4x x 19. 3 26 10 5 0x x x 20. 4 281x x

Solve each equation.

21. 4 22 26 28 0x x 22. 5 3 12x x x 23. 4 23 18 21x x

3 3 3

3 3 3

3

1. x + 64 2. x + 216 3. x -1000

4. x - 343 5. 8x -1 6. 8x - 125

7. 27x - 8 3 3

3 3 3

8. 8x +1000 9. 27x + 512

10. 64x + 27 11. 1000x -1 12. 125x + 64

4

Graphing Polynomial Equations

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

2 31. ( 2) ( 3)y x x x D 6 Root 0 -2 3

Mult. 1 2 3

2 2 32. ( 2) ( 3)y x x x Root

Mult.

23. ( 5) ( 1)y x x Root

Mult.

2 24. ( 5) ( 1)y x x Root

Mult.

35. ( 1) ( 1)( 2)y x x x Root

Mult.

36. ( 1) ( 1)( 2)y x x x Root

Mult.

2 27. ( 4) ( 3)y x x Root

Mult.

2 28. ( 4) ( 3)y x x Root

Mult.

2 39. ( 2) ( 3)y x x x Root

Mult.

2 3 210. ( 2) ( 3)y x x x Root

Mult.

A B

C

5

Graphing Polynomial Functions(continued)

E D F

G H

I

J

6

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. 3 211. ( ) 3 4 5 1; x 2p x x x x

Part 5: Application

12. A box is to be mailed. The volume in cubic inches of the box can be expressed as the

product of its three dimensions: 3 2( ) 16 79 120V x x x x . 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?

7

Honors Algebra 2 ~ Finding All Zeros Homework

I. 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, 4 3 2( ) 5 28 188 240f x x x x x , 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.

8

Honors Algebra 2 ~ Roots and Zeros Homework

I. 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

9

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 2006

On-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 Enrolled

2000 50

2004 65

2008 94

2010 110

Homework: Day 9

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

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

10

Honors Algebra 2 ~ Review for Unit 4 Test

1. Find the remainder for (x3 + 2x2 – 4x – 5) (x – 2)

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.

11

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 3

2. x = -3, x = -2, x = 5

3. 3 1x x

4. 2 9x x 5. 7, 5x x i

6. 3 22 16 32f x x x x

7. 3 211 23 45f x x x x

8. 3 2 15f x x x x

9. 3 134f

10. 5 3076f

11. 15, 3x x i

12. 3 17 2

, 4 3

x x

13. 3 4

, , 22 3

ix x x

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

2 2y x x degree 4

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

1 4y x x , degree 3

15. cubic: 3 2 2.013 .2 .52 1.40, .99398y x x x r

quartic: 4 3 2 2.003 .133 1.72 6.98 10.74, .998933y x x x x r

quartic is the best fit

16.84 million in 1994

16. 5 4 3 2 2 3 4 51024 3840 5760 4320 1620 243x x y x y x y xy y

17. 4 3 2 2 3 416 160 600 1000 625x x y x y xy y

12

Unit 4 Cumulative Review Questions

___________1. Solve for X.

8

11

6

121

231

200

X

A.

3

2

1

B.

36

43

16

C.

3

4.2

8.9

D.

8

23

16

___________2. Write the quadratic equation in standard form of a parabola containing the following points:

(-3,-4), (-1,0) and (9.-10)

__________3. Simplify i

i

32

21.

A. 7

8 i B.

13

74 i C.

7

78 i D. i74

__________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. 222 B. -94 C. 88 D. 2

__________7. Solve x2 + 8x = 6.

A. 114 B. 104 C. 228 D. 224

____________8. Factor completely: 643y

A. 3

4y B. 1644 2yyy C.

244 yy D. 64168 2

yyy

___________9. Solve x2 – x – 12 0.

A. [-4,3] B. [-3,4] C. ),4[]3,( D. ),3[}4,(

_____________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

13

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

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

3213422 23

xxxx B. 822 23

xxx C. 2

3222 23

xxxx D.

2

822 23

xxxx

____________15. Divide using long division: ).12()8824( 23xxxx

A. 12

1242 2

xx B.

12

942 2

xx C.

12

13522 2

xxx D.

12

14642

xxx

___________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: 4 23 54 0x x

A. -3 B. 3 C. -3i D. 6i

__________19. Which translation takes 2 1y x to 2y x ?

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