Warm-Up 5/3/13

Homework: Review 3.1-3.2 (due Mon) HW 3.3A #1-15 odds (due Tues)

Find the zeros and tell if the graph touches or crosses the x-axis. Tell the end behaviors.

1) f(x) = 2(x-5)(x+4)2

2) f(x) = 5x3 +7x2 -x +9 Use your calculator!

Answers:

1. x=5 mult=1, crosses x-axis , x=-4 mult= 2 touches x-axis and turns around. falls to the left and rises to the right. (odd exp)

2. Falls to the left and rises to the right.

Homework Answers: Pg. 323 (2-32 even)

2. Polynomial function, degree 44. Polynomial function, degree 76. Not a Polynomial function8. Not a Polynomial function10. Polynomial function, degree 212. Not a Polynomial function because graph is not smooth.14. Polynomial function16. C 18. d 20. f(x) = 11x3 -6x2 + x + 3; graph falls left and rises to the right. (odd) 22. f(x) = 11x4 -6x2 + x + 3; graph rises to the left and to the right. (even)24. f(x) = -11x4 -6x2 + x + 3; graph falls left and to the right. (Even and

neg)26. f(x) = 3(x+5)(x+2)2 x = -5 has multiplicity 1; The graph crosses the x-axis. x = -2 has multiplicity of 2; The graph touches the x-axis and turns

around.

Homework Answers cont: Pg. 323 (2-32 even)

28. f(x) = -3(x + ½)(x-4)3; x = -1/2 has multiplicity 1; Graph crosses the x-axis. x = 4 has multiplicity 3; graph crosses the x-axis

30. f(x) = x3+4x2+4x; x(x+2)2; x = 0 has multiplicity 1; Graph crosses the x-axis. x = -2 has multiplicity 2; graph touches the x-axis and turns around.

32. f(x) = x3+5x2-9x-45; (x-3)(x+3)(x+5); x = 3,-3,-5 have multiplicity 1; Graph crosses

the x-axis.

Announcements:

Quiz on Monday, May 6th

Lesson 3.3 Objective: Be able to use long and synthetic division to divide polynomials, evaluate a polynomial by using the Remainder Theorem, and solve a polynomial Equation by using the

Factor Theorem.

Lesson 3.3 Dividing Polynomials: Remainder and Factor Theorems

Long Division of Polynomials and the Division Algorithm

EXAMPLE 1: Divide: 2 14 45 9x x x

29 14 45x x x x

X2 + 9x--- ---

5x + 45

+ 5

5x + 45--- ---

0

= x + 5

Example 2: Divide

4 3 26 5 3 5 3 2x x x x x

2 4 3 23 2 6 5 0 3 5x x x x x x

Answer:

2

2 4 3 2

2 3 23 2 6 5 0 3 5

x xx x x x x x

Remainder: 7x-5/3x2 – 2x

You try: Use long division to divide.

2 4 32 2 3 7 10x x x x x

2x2 + 7x + 14 + 21x-10 x2-2x

Answer:

Synthetic Division: Shortcut to dividing polynomials of c =1. Example 3:

Synthetic Division cont:

You try: Use synthetic division to divide.

35 6 8 2x x x

Answer:

5x2 - 10x + 26 - 44

x + 2

The Factor Theorem:

Let f(x) be a polynomial.

a. If f(c) =0, then x-c is a factor of f(x).

b. If x-c is a factor of f(x), then f(c) = 0.

Example 4: Solve the equation 2x3 – 3x2-11x + 6 = 0 given that 3 is a zero of f(x) = 2x3 -3x2-11x + 6.

Step 1: Use synthetic division and solve.

3| 2 -3 -11 6

6 9 -6

2 3 -2 0

The remainder is 0, which means that x-3 is a factor of

2x3 -3x2-11x + 6

What are the factors of 2x3 -3x2-11x + 6?

Using synthetic division, we found the factors to be (x-3)(2x2+ 3x-2) =0.

Finish factoring: (x-3) (2x-1) (x+2) =0

X=3, x=1/2, x=-2•Find the x- intercepts.

• The solution set is {-2, 1/2, 3}

You try: f(x)= 15x3+14x2 -3x -2=0, given that -1 is a zero of f(x), find all factors.

Answers: -1| 15 14 -3 -2

-15 1 2

15 -1 -2 0

Factors: (x+1) (15x2 –x -2)=0

(x+1) (5x-2)(3x+1)=0

x = -1, x= 2/5, x = -1/3

Solution Set {-1, -1/3, 2/5}

Based on the Factor Theorem, the following statements are useful in solving polynomial

equations.

Summary:

Explain how the Factor Theorem can be used to determine if x-1 is a factor of

x3 -2x2 – 11x + 12.