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LESSON 5.6 Rational Zeros of Polynomial Functions.

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1 LESSON 5.6 Rational Zeros of Polynomial Functions

2 Solve x 3 – 5x 2 + 8x – 4= 0. How? Can you factor this? Can you use the quadratic formula? Now what if I tell you that one root is x =1? Then x 3 – 5x 2 + 8x – 4 = (x – 1)( ). Use synthetic division to find the missing factor. The missing factor is x 2 – 4x + 4, which can be factored. So solve and learn that x = 2. So the solutions are 1 and 2 (double root).

3 But what if you don’t know any zeros and cannot factor a polynomial? Rational Zero Theorem To find the possible rational zeros of a polynomial divide the factors of the constant by the factors of the leading coefficient. List the possible rational zeros for f(x) = 2x 3 + 3x 2 – 4x + 6. Answer: ±1,±2,±3,±6,±1/2,±3/2

4 Now try this one…. a)Solve x 3 + 4x 2 + x – 6 = 0 Answer: x = 1, x = -2, x = -3 Do they all check? Why do you think there were 3 solutions?

5 and this one… b) Find all of the roots of f(x) = x 4 + 2x 3 – 7x 2 – 20x – 12 Answer: x = 3, -2 (double root) and -1. Verify this using your graphing calculator.

6 and one last problem! c) Find the zeros of f(x) = 6x 4 – 25x 3 + 30x 2 – 5x – 6 How many answers should there be? Answer: x = 1, x = 2, x = 3/2, x = -1/3

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