1. 7.6 Rational Zero Theorem Objectives: 1. Identify the possible rational zeros of a polynomial function. 2. Find all the rational zeros of a polynomial function.

2. Rational Zero Theorem and Corollary Let f(x) = a 0 x n + a 1 x n-1 +…+a n-1 x + a n represent a polynomial function with integral coefficients. If p / q is a rational number in simplest form and is a zero of y = f(x), then p is a factor of a n and q is a factor of a 0. Example: Given 4x3 + 3x 2 – 7x + 18. If 9 / 2 is a zero of f(x), then 9 is a factor of 18 and 2 is a factor of 4. Corollary (Integral Zero Theorem) If the coefficients of a polynomial function are integers such that a 0 = 1 and a n ≠ 0, any rational zeros of the function must be factors of a n.

3. In terms you can understand! Given f(x) = x 4 + 6x 3 + 7x 2 – 6x – 8. Find the possible rational roots. The only rational roots are factors of 8 (a n ) divided by factors of 1(a 0 ). a n = -8. so the factors are ±1, ±2, ±4, ±8 a 0 = 1, so the factors are ± 1. So if f(x) HAS rational roots, the only ones can be ±1, ±2, ±4, ± 8

4. Another problem Find the possible rational roots of f(x) = 6x 4 + 2x 3 – x 2 + 7x – 4 a n = -4,so factors are ±1, ±2, ±4 a 0 = 6, so factors are ±1, ±2, ±3, ±6 So all possible rational zeros are the factors of a n : ±1, ±2, ±4, and p / q : ± 1 / 2, ± 1 / 3, ± 1 / 6, ± 2 / 3, ± 4 / 3

5. Find zeros Given f(x) = 2x 4 – 5x 3 – 4x 2 + 10x – 3, find all the zeros. Using Descarteś Rule of signs: Positive real roots, 3 or 1. f(-x) = 2x 4 + 5x 3 – 4x 2 – 10x – 3 Negative real roots, 1. a 0 = 2, factors:  1,  2 a n = -3, factors;  1,  3 So my possible rational roots are ±1, ±3, ± 1 / 2, ± 3 / 2

6. continued f(x) = 2x 4 – 5x 3 – 4x 2 + 10x – 3 we know to test ±1, ±3, ± 1 / 2, ± 3 / 2 using synthetic division 2x³-3x²-7x+3 Start over X2-5-410-3 2-737-10 no 12-3-730 yes x2-3-73 -32-920-57 no 32323 no ½2-2-81 no -½2-4-55.5 no 3/220-7-7.5 -3/22-620

7. continued Solve remaining polynomial 2x²-6x+2=0 Roots are: 1, -3/2,

8. Find all zeros. h(x) = 10x 3 – 17x 2 – 7x + 2 D ROS gives us 2 or 0 positive real roots and 1 negative real root. Our possible roots are ±1, ±2, ± 1 / 2, ± 1 / 5, ± 1 / 10, ± 2 / 5 We either have 2 positive and 1 negative real roots or 2 imaginary and 1 negative real. Find out what they are!

9. Homework p. 381 12-32 even