Section 9-5
 The Factor Theorem
Warm-up
                                  2
                 Let f(x) = 3x − 40x + 48.
     1. Which of the following poly...
Warm-up
                                 2
                Let f(x) = 3x − 40x + 48.
    1. Which of the following polynom...
Warm-up
                            2
            Let f(x) = 3x − 40x + 48.
1. Which of the following polynomials is a fac...
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 o...
Factor-Solution-Intercept Equivalence
Theorem
Factor-Solution-Intercept Equivalence
Theorem
For any polynomial f, the following are all equivalent:
Factor-Solution-Intercept Equivalence
Theorem
For any polynomial f, the following are all equivalent:

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

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

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

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

(x - c) is a factor...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 1
Factor by finding one zero of the polynomial using your graphing
          calculator and then dividing to find th...
Example 2
Find an equation for a polynomial function p with the zeros 2, -4,
                            and 4/7.
Example 2
Find an equation for a polynomial function p with the zeros 2, -4,
                            and 4/7.

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

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

       ...
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)...
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)...
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)...
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)...
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)...
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)...
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)...
Homework
Homework

       p. 587 #2-18
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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

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