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)
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