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# 9-5 Notes

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The Factor Theorem

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### 9-5 Notes

1. 1. Section 9-5 The Factor Theorem
2. 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. 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. 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. 5. Solver
6. 6. Solver
7. 7. Solver
8. 8. Solver
9. 9. Solver
10. 10. Factor Theorem
11. 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. 12. Factor-Solution-Intercept Equivalence Theorem
13. 13. Factor-Solution-Intercept Equivalence Theorem For any polynomial f, the following are all equivalent:
14. 14. Factor-Solution-Intercept Equivalence Theorem For any polynomial f, the following are all equivalent: (x - c) is a factor of f
15. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 33. Example 2 Find an equation for a polynomial function p with the zeros 2, -4, and 4/7.
34. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 44. Homework
45. 45. Homework p. 587 #2-18