1. Mrs. McConaughy Honors Algebra 2 1 Lesson 6.5: Finding Rational Zeros Objective: To find the rational zeros of a polynomial function

2. Mrs. McConaughy Honors Algebra 2 2 Before we start, recall: A rational number is one that can be written ________________________ ______________________________. The real-number zeros of a polynomial function are either ________ or __________. as the quotient (ratio) of two integers. rationalirrational

3. Mrs. McConaughy Honors Algebra 2 3 EXAMPLES Find the zeros of the following functions: f(x) = x 2 - 3x – 4 f(x) = x 2 - 2

4. Mrs. McConaughy Honors Algebra 2 4 The Rational Zero Test Consider the polynomial f(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0 with integer coefficients. Every rational zero of f has the form p = ______________________ q factors of constant term factors of leading coefficient

5. Mrs. McConaughy Honors Algebra 2 5 EXAMPLE 1 Find the rational zeros of the following function: f(x) = x 4 - x 3 – 5x 2 + 3x + 6 STEP 1: Find all possible rational zeros: ------------- STEP 2: Use the Remainder Theorem and Synthetic Division to determine actual zeros: f(x) = ________________ ________________________________ (x-1) (x-2)(x 2 -3)

6. Mrs. McConaughy Honors Algebra 2 6 From the previous factorization, you can see that f has only ______ rational zeros: ______________ (The other two zeros, ______, are irrational.) two f(x) = x 4 -x 3 –5x 2 +3x+6 NOTE: Graphing can speed up the process of finding the first zero. x = – 1 and x = 2.

7. Mrs. McConaughy Honors Algebra 2 7 EXAMPLE 2 Find the real zeros of the following function: f(x) = 10x 3 – 15x 2 - 16x + 12 STEP 1: Find all possible rational zeros: ------------ STEP 2: Use the Remainder Theorem and Synthetic Division to determine actual zeros. Shorten your search for the initial rational zero by graphing on a graphics calculator, first. graphing f(x) = ___________________________

8. Mrs. McConaughy Honors Algebra 2 8 The Fundamental Theorem of Algebra Counting complex and repeated solutions, an nth-degree polynomial equation has exactly n solutions. Carl Friedrich Gauss

9. Mrs. McConaughy Honors Algebra 2 9 FINAL CHECKS FOR UNDERSTANDING 1.List the possible rational zeros given by the Rational-Zero Test for: f(x) = 3x 3 – 6x 2 + 7x - 2 2. Factor x 3 – 3x 2 - 6x + 8 completely. 3. Factor x 3 +4x 2 - x - 4 completely. 4.Which is the more efficient method, factoring or rational-zero test, for finding all real zeros of the function, g(x) = x 3 + x 2 -2x – 2? Explain.

10. Mrs. McConaughy Honors Algebra 2 10 In 4-6, find all real zeros of the polynomial function. y = 4x 3 – 12x 2 - x + 15y = -4x 3 + 15x 2 - 8x - 3y = -3x 3 + 20x 2 - 36x + 16 HOMEWORK ASSIGNMENT: ______________________________

11. Mrs. McConaughy Honors Algebra 2 11 Click to return to lesson. Three reasonable choices for x would be: _____________ -6/5, 1/2, 2 f(x) = _________________________