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2.5 Zeros of Polynomial Functions

2.5 Zeros of Polynomial Functions. Fundamental Theorem of Algebra Rational Zero Test Upper and Lower bound Rule. Fundamental Theorem of Algebra. If f(x) is a polynomial of degree “n” > 0, then f(x) has at least one zero in the complex number system. Complex zero’s (roots) come in pairs

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2.5 Zeros of Polynomial Functions

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  1. 2.5 Zeros of Polynomial Functions Fundamental Theorem of Algebra Rational Zero Test Upper and Lower bound Rule

  2. Fundamental Theorem of Algebra If f(x) is a polynomial of degree “n” > 0, then f(x) has at least one zero in the complex number system. Complex zero’s (roots) come in pairs If a + bi is a zero, then a – bi is a zero.

  3. Linear Factorization Theorem If f(x) is a polynomial of degree “n”>0, then there are as many zeros as degree. If f(x) is a third degree function, then f(x) = an(x – c1)(x – c2)(x – c3) where care complex numbers. Complex zero’s (roots) come in pairs If a + bi is a zero, then a – bi is a zero.

  4. The Rational Zero Test If f(x) has integer coefficients, then all possible zeros are factors of the constant factor of the lead coefficient

  5. The Rational Zero Test If f(x) has integer coefficients, then all possible zeros are factors of the constant factor of the lead coefficient f(x) = x 3 – 7x 2 + 4x + 12 Possible zeros ± 1, ± 2, ± 3, ± 4, ± 6, ± 12 ± 1

  6. f(x) = x 3 – 7x 2 + 4x + 12Possible zeros ± 1, ± 2, ± 3, ± 4, ± 6, ± 12 ± 1 - 1 | 1 -7 4 12 -1 8 -12 1 - 8 12 0 So – 1 is a zero How do you want to find the other zeros. x 2 – 8x + 12

  7. Find the zeros f(x) = 3x3 – x2 + 6x - 2

  8. Descartes' Rule of Signs Let f(x) = anxn + an-1xn-1 + ….a1x + a0 ; with real coefficients and a0≠ 0. Part 1 The number of positive real zeros equals (or a even number less), the number of variation in the sign of the coefficient (switching from positive to negative or negative to positive).

  9. Descartes' Rule of Signs Let f(x) = anxn + an-1xn-1 + ….a1x + a0 ; with real coefficients and a0≠ 0. Part 2 The number of negative real zeros equals (or a even number less), the number of variation in the sign of the coefficient (switching from positive to negative or negative to positive) in f(- x).

  10. Using the Desecrate rule of signs f(x) = 4x3 - 3x2 +2x – 1 How many times does the sign change ?

  11. Using the Desecrate rule of signs f(x) = 4x3 - 3x2 +2x – 1 How many times does the sign change ? 3 times. There are 3 or 1 positive zeros.

  12. Using the Desecrate rule of signs f(x) = 4x3 - 3x2 +2x – 1 What about f( -x) = -4x3 – 3x2 – 2x - 1 How many times does the sign change ?

  13. Using the Desecrate rule of signs f(x) = 4x3 - 3x2 +2x – 1 What about f( -x) = -4x3 – 3x2 – 2x - 1 How many times does the sign change ? No change, no negative zeros.

  14. Upper and Lower bound Rule If c > 0 ( “c” the number you divide by) and the last row of synthetic division is all positive or zero, the c| is the upper bound So there is no zero larger then c, where c > 0. If c < 0 and the last row alternate signs ( zero count either way), then c is the lower bound.

  15. f(x) = 2x3 – 5x2 + 12x - 5 Check to see if 3 is the upper bound? 3| 2 - 5 12 - 5 All signs are 6 3 45 positive. 2 1 15 40 3 is an upper bound

  16. f(x) = 2x3 – 5x2 + 12x - 5 Check to see if - 1 is the lower bound? - 1| 2 - 5 12 - 5 All signs are -2 7 -19 switch. 2 - 7 19 -24 -1 is an lower bound

  17. f(x) = 2x3 – 5x2 + 12x - 5 Find the zeros

  18. Homework Page 160 – 164 # 5, 15, 23, 35, 42, 50, 57, 65, 73, 81, 85, 93, 103, 108, 111

  19. Homework Page 160 – 164 # 9, 19, 29, 41, 53, 61, 64, 77, 87, 97, 105,125

  20. One more time • http://www.youtube.com/watch?v=VK8qDdeLtsw&feature=related

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