3.3 Real Zeros of         PolynomialsPhilippians 4:6-7 do not be anxious aboutanything, but in everything by prayer andsup...
Rational Zeros Theorem
Rational Zeros TheoremIf P(x) = an x + an−1 x              n           n−1                                + an−2 x   n−2  ...
Find all rational zeros of P(x) = x − 11x + 23x + 35                                  3     2
Find all rational zeros of P(x) = x − 11x + 23x + 35                                  3     2               p   1 5 7 35  ...
Find all rational zeros of P(x) = x − 11x + 23x + 35                                  3     2               p   1 5 7 35  ...
Find all rational zeros of P(x) = x − 11x + 23x + 35                                  3     2               p   1 5 7 35  ...
Find all rational zeros of P(x) = x − 11x + 23x + 35                                  3     2               p   1 5 7 35  ...
Find all rational zeros of P(x) = x − 11x + 23x + 35                                   3    2               p   1 5 7 35  ...
Factor 3x − 4x − 13x − 6         3    2
Factor 3x − 4x − 13x − 6               3     2This means we are looking for the zeros.
Factor 3x − 4x − 13x − 6                3     2 This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± , , , , , ...
Factor 3x − 4x − 13x − 6                3     2 This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± , , , , , ...
Factor 3x − 4x − 13x − 6                3     2 This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± , , , , , ...
Factor 3x − 4x − 13x − 6                      3    2  This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± , , ...
Factor 3x − 4x − 13x − 6                      3       2  This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± ,...
Factor 3x − 4x − 13x − 6                      3       2  This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± ,...
Factor 3x − 4x − 13x − 6                      3       2  This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± ,...
Factor 3x − 4x − 13x − 6                      3       2  This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± ,...
Factor 3x − 4x − 13x − 6                      3       2  This means we are looking for the zeros.p   1 2 3 6 1 2 3 6  =± ,...
3Find the exact zeros of f (x) = x − 6x + 4
3Find the exact zeros of f (x) = x − 6x + 4    p      = ± 1, 2, 4   standard window    q
3Find the exact zeros of f (x) = x − 6x + 4    p      = ± 1, 2, 4    standard window    q              graph and test
3Find the exact zeros of f (x) = x − 6x + 4    p      = ± 1, 2, 4    standard window    q              graph and test    x=2
3Find the exact zeros of f (x) = x − 6x + 4    p      = ± 1, 2, 4    standard window    q              graph and test    x...
3Find the exact zeros of f (x) = x − 6x + 4    p      = ± 1, 2, 4    standard window    q              graph and test    x...
3Find the exact zeros of f (x) = x − 6x + 4    p      = ± 1, 2, 4    standard window    q              graph and test    x...
3Find the exact zeros of f (x) = x − 6x + 4
3Find the exact zeros of f (x) = x − 6x + 4                             2       1 0 -6 4                                  ...
3Find the exact zeros of f (x) = x − 6x + 4  2 x + 2x − 2                  2       1 0 -6 4                               ...
3  Find the exact zeros of f (x) = x − 6x + 4    2   x + 2x − 2                  2       1 0 -6 4   −2 ± 4 − (4)(−2)      ...
3  Find the exact zeros of f (x) = x − 6x + 4    2   x + 2x − 2                  2       1 0 -6 4   −2 ± 4 − (4)(−2)      ...
3  Find the exact zeros of f (x) = x − 6x + 4    2   x + 2x − 2                  2       1 0 -6 4   −2 ± 4 − (4)(−2)      ...
3  Find the exact zeros of f (x) = x − 6x + 4     2   x + 2x − 2                  2       1 0 -6 4   −2 ± 4 − (4)(−2)     ...
3  Find the exact zeros of f (x) = x − 6x + 4     2   x + 2x − 2                  2       1 0 -6 4   −2 ± 4 − (4)(−2)     ...
Find the exact zeros of P(x) = x + 4x + 3x − 2                                3    2
Find the exact zeros of P(x) = x + 4x + 3x − 2                                  3   2                x = −2, − 1 ± 2
4   2Find all real zeros of f (x) = 10x − x + 4x − 6
4   2Find all real zeros of f (x) = 10x − x + 4x − 6  Doesn’t say exact ... approximations OK!
4   2Find all real zeros of f (x) = 10x − x + 4x − 6  Doesn’t say exact ... approximations OK!      p        → [ −6,6 ]   ...
4   2Find all real zeros of f (x) = 10x − x + 4x − 6  Doesn’t say exact ... approximations OK!      p        → [ −6,6 ]   ...
4   2Find all real zeros of f (x) = 10x − x + 4x − 6  Doesn’t say exact ... approximations OK!      p        → [ −6,6 ]   ...
HW #3“Never doubt that a small group of thoughtfulcommitted people can change the world; indeedit is the only thing that e...
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0303 ch 3 day 3

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  • 0303 ch 3 day 3

    1. 1. 3.3 Real Zeros of PolynomialsPhilippians 4:6-7 do not be anxious aboutanything, but in everything by prayer andsupplication with thanksgiving let your requestsbe made known to God. And the peace of God,which surpasses all understanding, will guardyour hearts and your minds in Christ Jesus.
    2. 2. Rational Zeros Theorem
    3. 3. Rational Zeros TheoremIf P(x) = an x + an−1 x n n−1 + an−2 x n−2 + ... + a1 x + a0has integral coefficients, then every rational zero pof P(x) is of the form where q p is a factor of the constant term, and q is a factor of the leading coefficient.
    4. 4. Find all rational zeros of P(x) = x − 11x + 23x + 35 3 2
    5. 5. Find all rational zeros of P(x) = x − 11x + 23x + 35 3 2 p 1 5 7 35 =± , , , q 1 1 1 1
    6. 6. Find all rational zeros of P(x) = x − 11x + 23x + 35 3 2 p 1 5 7 35 =± , , , q 1 1 1 1 set the window on your grapher to [-35,35]
    7. 7. Find all rational zeros of P(x) = x − 11x + 23x + 35 3 2 p 1 5 7 35 =± , , , q 1 1 1 1 set the window on your grapher to [-35,35] graph and test
    8. 8. Find all rational zeros of P(x) = x − 11x + 23x + 35 3 2 p 1 5 7 35 =± , , , q 1 1 1 1 set the window on your grapher to [-35,35] graph and test (we are applying the Remainder Theorem here)
    9. 9. Find all rational zeros of P(x) = x − 11x + 23x + 35 3 2 p 1 5 7 35 =± , , , q 1 1 1 1 set the window on your grapher to [-35,35] graph and test (we are applying the Remainder Theorem here) x = − 1, 5, 7
    10. 10. Factor 3x − 4x − 13x − 6 3 2
    11. 11. Factor 3x − 4x − 13x − 6 3 2This means we are looking for the zeros.
    12. 12. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , ,q 1 1 1 1 3 3 3 3
    13. 13. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , , window: [-6,6]q 1 1 1 1 3 3 3 3
    14. 14. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , , window: [-6,6]q 1 1 1 1 3 3 3 3 2 Zeros are: −1, − , 3 3
    15. 15. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , , window: [-6,6]q 1 1 1 1 3 3 3 3 2 Zeros are: −1, − , 3 2 3x=− 33x = −23x + 2 = 0
    16. 16. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , , window: [-6,6]q 1 1 1 1 3 3 3 3 2 Zeros are: −1, − , 3 2 3x=− 3 x = −13x = −23x + 2 = 0 x +1 = 0
    17. 17. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , , window: [-6,6]q 1 1 1 1 3 3 3 3 2 Zeros are: −1, − , 3 2 3x=− 3 x = −1 x=33x = −23x + 2 = 0 x +1 = 0 x−3= 0
    18. 18. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , , window: [-6,6]q 1 1 1 1 3 3 3 3 2 Zeros are: −1, − , 3 2 3x=− 3 x = −1 x=33x = −23x + 2 = 0 x +1 = 0 x−3= 0 (3x + 2)(x + 1)(x − 3)
    19. 19. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , , window: [-6,6]q 1 1 1 1 3 3 3 3 2 Zeros are: −1, − , 3 2 3x=− 3 x = −1 x=33x = −23x + 2 = 0 x +1 = 0 x−3= 0 ⎛ 2 ⎞ (3x + 2)(x + 1)(x − 3) Do not use ⎜ x + ⎟ ⎝ 3 ⎠
    20. 20. Factor 3x − 4x − 13x − 6 3 2 This means we are looking for the zeros.p 1 2 3 6 1 2 3 6 =± , , , , , , , window: [-6,6]q 1 1 1 1 3 3 3 3 2 Zeros are: −1, − , 3 2 3x=− 3 x = −1 x=33x = −23x + 2 = 0 x +1 = 0 x−3= 0 ⎛ 2 ⎞ (3x + 2)(x + 1)(x − 3) Do not use ⎜ x + ⎟ ⎝ 3 ⎠
    21. 21. 3Find the exact zeros of f (x) = x − 6x + 4
    22. 22. 3Find the exact zeros of f (x) = x − 6x + 4 p = ± 1, 2, 4 standard window q
    23. 23. 3Find the exact zeros of f (x) = x − 6x + 4 p = ± 1, 2, 4 standard window q graph and test
    24. 24. 3Find the exact zeros of f (x) = x − 6x + 4 p = ± 1, 2, 4 standard window q graph and test x=2
    25. 25. 3Find the exact zeros of f (x) = x − 6x + 4 p = ± 1, 2, 4 standard window q graph and test x=2 but then the other 2 roots must be irrational
    26. 26. 3Find the exact zeros of f (x) = x − 6x + 4 p = ± 1, 2, 4 standard window q graph and test x=2 but then the other 2 roots must be irrational “exact zeros” ... no calculator!
    27. 27. 3Find the exact zeros of f (x) = x − 6x + 4 p = ± 1, 2, 4 standard window q graph and test x=2 but then the other 2 roots must be irrational “exact zeros” ... no calculator!use synthetic division until it’s a quadratic then use the Quadratic Formula
    28. 28. 3Find the exact zeros of f (x) = x − 6x + 4
    29. 29. 3Find the exact zeros of f (x) = x − 6x + 4 2 1 0 -6 4 2 4 -4 1 2 -2 0
    30. 30. 3Find the exact zeros of f (x) = x − 6x + 4 2 x + 2x − 2 2 1 0 -6 4 2 4 -4 1 2 -2 0
    31. 31. 3 Find the exact zeros of f (x) = x − 6x + 4 2 x + 2x − 2 2 1 0 -6 4 −2 ± 4 − (4)(−2) 2 4 -4x= 2 1 2 -2 0
    32. 32. 3 Find the exact zeros of f (x) = x − 6x + 4 2 x + 2x − 2 2 1 0 -6 4 −2 ± 4 − (4)(−2) 2 4 -4x= 2 1 2 -2 0 −2 ± 12x= 2
    33. 33. 3 Find the exact zeros of f (x) = x − 6x + 4 2 x + 2x − 2 2 1 0 -6 4 −2 ± 4 − (4)(−2) 2 4 -4x= 2 1 2 -2 0 −2 ± 12x= 2 −2 ± 2 3x= 2
    34. 34. 3 Find the exact zeros of f (x) = x − 6x + 4 2 x + 2x − 2 2 1 0 -6 4 −2 ± 4 − (4)(−2) 2 4 -4x= 2 1 2 -2 0 −2 ± 12x= 2 −2 ± 2 3x= 2x = −1 ± 3
    35. 35. 3 Find the exact zeros of f (x) = x − 6x + 4 2 x + 2x − 2 2 1 0 -6 4 −2 ± 4 − (4)(−2) 2 4 -4x= 2 1 2 -2 0 −2 ± 12x= 2 x = 2, − 1 ± 3 −2 ± 2 3x= 2x = −1 ± 3
    36. 36. Find the exact zeros of P(x) = x + 4x + 3x − 2 3 2
    37. 37. Find the exact zeros of P(x) = x + 4x + 3x − 2 3 2 x = −2, − 1 ± 2
    38. 38. 4 2Find all real zeros of f (x) = 10x − x + 4x − 6
    39. 39. 4 2Find all real zeros of f (x) = 10x − x + 4x − 6 Doesn’t say exact ... approximations OK!
    40. 40. 4 2Find all real zeros of f (x) = 10x − x + 4x − 6 Doesn’t say exact ... approximations OK! p → [ −6,6 ] standard window q
    41. 41. 4 2Find all real zeros of f (x) = 10x − x + 4x − 6 Doesn’t say exact ... approximations OK! p → [ −6,6 ] standard window q graphing suggests 2 zeros ... they are:
    42. 42. 4 2Find all real zeros of f (x) = 10x − x + 4x − 6 Doesn’t say exact ... approximations OK! p → [ −6,6 ] standard window q graphing suggests 2 zeros ... they are: x ≈ −1.03, .77 and the other two are imaginary
    43. 43. HW #3“Never doubt that a small group of thoughtfulcommitted people can change the world; indeedit is the only thing that ever has.” Margaret Mead

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