The document discusses using the factor theorem to factor polynomials. It provides examples of finding the factors of a polynomial given its zeros. It also presents the factor-solution-intercept equivalence theorem, which states that for any polynomial f, the following are equivalent: (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, and the remainder when f(x) is divided by (x - c) is 0. Examples are worked through to demonstrate factoring polynomials by finding zeros and dividing.

1. Section 9-5
The Factor Theorem

2. Warm-up
2
Let f(x) = 3x − 40x + 48.
1. Which of the following polynomials is a factor of f(x)?
a. x − 2 b. x − 3 c. x − 4 d. x − 6 e. x − 12 f. x − 24
2. Which of the given values equals 0?
a. f(2) b. f(3) c. f(4) d. f(6) e. f(12) f. f(24)

3. Warm-up
2
Let f(x) = 3x − 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?
a. f(2) b. f(3) c. f(4) d. f(6) e. f(12) f. f(24)

4. Warm-up
2
Let f(x) = 3x − 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)

5. Solver

6. Solver

7. Solver

8. Solver

9. Solver

10. Factor Theorem

11. 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.

12. Factor-Solution-Intercept Equivalence
Theorem

13. Factor-Solution-Intercept Equivalence
Theorem
For any polynomial f, the following are all equivalent:

14. Factor-Solution-Intercept Equivalence
Theorem
For any polynomial f, the following are all equivalent:
(x - c) is a factor of f

15. Factor-Solution-Intercept Equivalence
Theorem
For any polynomial f, the following are all equivalent:
(x - c) is a factor of f
f(c) = 0

16. Factor-Solution-Intercept Equivalence
Theorem
For any polynomial f, the following are all equivalent:
(x - c) is a factor of f
f(c) = 0
c is an x-intercept of the graph y = f(x)

17. Factor-Solution-Intercept Equivalence
Theorem
For any polynomial f, the following are all equivalent:
(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

18. Factor-Solution-Intercept Equivalence
Theorem
For any polynomial f, the following are all equivalent:
(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

19. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6

20. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6

21. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
4x + 112x 3 − 41x 2 + 13x + 6

22. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x
4x + 112x 3 − 41x 2 + 13x + 6

23. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )

24. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
−44x 2 + 13x

25. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x −11x
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
−44x 2 + 13x

26. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x −11x
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
−44x 2 + 13x
−(−44x 2 − 11x)

27. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x −11x
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
−44x 2 + 13x
−(−44x 2 − 11x)
24x + 6

28. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x −11x +6
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
−44x 2 + 13x
−(−44x 2 − 11x)
24x + 6

29. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x −11x +6
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
−44x 2 + 13x
−(−44x 2 − 11x)
24x + 6
−(24x + 6)

30. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x −11x +6
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
2
−44x 2 + 13x 3x − 11x + 6
−(−44x 2 − 11x)
24x + 6
−(24x + 6)

31. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x −11x +6
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
2
−44x 2 + 13x 3x − 11x + 6
−(−44x 2 − 11x)
(3x − 2)(x − 3)
24x + 6
−(24x + 6)

32. Example 1
Factor by finding one zero of the polynomial using your graphing
calculator and then dividing to find the others.
3 2
12x − 41x + 13x + 6
2
3x −11x +6
4x + 112x 3 − 41x 2 + 13x + 6
3 2
−(12x + 3x )
2
−44x 2 + 13x 3x − 11x + 6
−(−44x 2 − 11x)
(3x − 2)(x − 3)
24x + 6
−(24x + 6)
Answer:
(4x + 1)(3x − 2)(x − 3)

33. Example 2
Find an equation for a polynomial function p with the zeros 2, -4,
and 4/7.

34. Example 2
Find an equation for a polynomial function p with the zeros 2, -4,
and 4/7.
p(x) = (x − 2)(x + 4)(7x − 4)

35. Example 2
Find an equation for a polynomial function p with the zeros 2, -4,
and 4/7.
p(x) = (x − 2)(x + 4)(7x − 4)
2
= (x + 2x − 8)(7x − 4)

36. Example 2
Find an equation for a polynomial function p with the zeros 2, -4,
and 4/7.
p(x) = (x − 2)(x + 4)(7x − 4)
2
= (x + 2x − 8)(7x − 4)
3 2
= 7x + 10x − 64x + 32

37. Example 3
Find four linear factors of a polynomial t(r) if t(-2) = 0, t(4) = 0,
t(6) = 0, and t(-4/3) = 0.

38. Example 3
Find 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

39. Example 3
Find 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

40. Example 3
Find 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

41. Example 3
Find 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+ 4
3

42. Example 3
Find 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+ 4
3

43. Example 3
Find 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+ 4
3
3t + 4

44. Homework

45. Homework
p. 587 #2-18