Section 9-5The Factor Theorem

Warm-up Let f(x) = 3x2 − 40x + 48.

1. Which of the following polynomials is a factor of f(x)?

e. x −12 a. x − 2 b. x − 3 c. x − 4 d. x − 6 f. x − 24

2. Which of the given values equals 0?

e. f(12) a. f(2) b. f(3) c. f(4) d. f(6) f. f(24)

Warm-up Let f(x) = 3x2 − 40x + 48.

1. Which of the following polynomials is a factor of f(x)?

e. x −12

2. Which of the given values equals 0?

e. f(12) a. f(2) b. f(3) c. f(4) d. f(6) f. f(24)

Warm-up Let f(x) = 3x2 − 40x + 48.

1. Which of the following polynomials is a factor of f(x)?

e. x −12

2. Which of the given values equals 0?

e. f(12)

Solver

Solver

Solver

Solver

Solver

Factor Theorem

Factor Theorem

For a polynomial f(x), a number c is a solution to f(x) = 0 IFF (x - c) is a factor of f.

Factor-Solution-Intercept Equivalence Theorem

Factor-Solution-Intercept Equivalence Theorem

For any polynomial f, the following are all equivalent:

Factor-Solution-Intercept Equivalence Theorem

(x - c) is a factor of f

For any polynomial f, the following are all equivalent:

Factor-Solution-Intercept Equivalence Theorem

(x - c) is a factor of f

f(c) = 0

For any polynomial f, the following are all equivalent:

Factor-Solution-Intercept Equivalence Theorem

(x - c) is a factor of f

f(c) = 0

c is an x-intercept of the graph y = f(x)

For any polynomial f, the following are all equivalent:

Factor-Solution-Intercept Equivalence Theorem

(x - c) is a factor of f

f(c) = 0

c is an x-intercept of the graph y = f(x)

c is a zero of f

For any polynomial f, the following are all equivalent:

Factor-Solution-Intercept Equivalence Theorem

(x - c) is a factor of f

f(c) = 0

c is an x-intercept of the graph y = f(x)

c is a zero of f

The remainder when f(x) is divided by (x - c) is 0

For any polynomial f, the following are all equivalent:

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others. 12x3 − 41x2 +13x + 6

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others. 12x3 − 41x2 +13x + 6

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

−11x

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

−11x

−(−44x2 −11x)

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

−11x

−(−44x2 −11x) 24x + 6

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

−11x

−(−44x2 −11x) 24x + 6

+6

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

−11x

−(−44x2 −11x) 24x + 6

+6

−(24x + 6)

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

−11x

−(−44x2 −11x) 24x + 6

+6

−(24x + 6)

3x2 −11x + 6

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

−11x

−(−44x2 −11x) 24x + 6

+6

−(24x + 6)

3x2 −11x + 6

(3x − 2)(x − 3)

Example 1Factor by finding one zero of the polynomial using your graphing

calculator and then dividing to find the others.

3x2

−(12x3 + 3x2) 4x +112x3 − 41x2 +13x + 6

12x3 − 41x2 +13x + 6

−44x2 +13x

−11x

−(−44x2 −11x) 24x + 6

+6

−(24x + 6)

3x2 −11x + 6

(3x − 2)(x − 3)

(4x +1)(3x − 2)(x − 3)Answer:

Example 2Find an equation for a polynomial function p with the zeros 2, -4,

and 4/7.

Example 2Find an equation for a polynomial function p with the zeros 2, -4,

and 4/7.

p(x) = (x − 2)(x + 4)(7x − 4)

Example 2Find an equation for a polynomial function p with the zeros 2, -4,

and 4/7.

p(x) = (x − 2)(x + 4)(7x − 4)

= (x2 + 2x − 8)(7x − 4)

Example 2Find an equation for a polynomial function p with the zeros 2, -4,

and 4/7.

p(x) = (x − 2)(x + 4)(7x − 4)

= (x2 + 2x − 8)(7x − 4)

= 7x3 +10x2 − 64x + 32

Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,

t(6) = 0, and t(-4/3) = 0.

Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,

t(6) = 0, and t(-4/3) = 0.

t + 2

Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,

t(6) = 0, and t(-4/3) = 0.

t + 2 t − 4

Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,

t(6) = 0, and t(-4/3) = 0.

t + 2 t − 4 t − 6

Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,

t(6) = 0, and t(-4/3) = 0.

t + 2 t − 4 t − 6

t +43

Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,

t(6) = 0, and t(-4/3) = 0.

t + 2 t − 4 t − 6

t +43

Example 3Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,

t(6) = 0, and t(-4/3) = 0.

t + 2 t − 4 t − 6

t +43 3t + 4

Homework

Homework

p. 587 #2-18