1. Table of Contents Polynomials: The Rational Zero Test The Rational Zero Test states that if a polynomial with real coefficients has rational zeros each will be on the following list:  c/d, where c is a factor of the constant term and d is a factor of the leading coefficient. Example: Make a list of possible rational zeros of the polynomial, P(x) = 4x 3 + 19x 2 + 20x + 6. The constant term is 6. Its factors are 1, 2, 3 and 6. These are values of c. The leading coefficient is 4. Its factors are 1, 2, and 4. These are values of d.

2. Table of Contents Polynomials: The Rational Zero Test Slide 2 c-values – 1, 2, 3 and 6 d-values – 1, 2 and 4 Now form all fractions of the form:  c/d: Simplifying and removing duplicate fractions results in:

3. Table of Contents Polynomials: The Rational Zero Test Slide 3 The polynomial in the example, (P(x) = 4x 3 + 19x 2 + 20x + 6) actually has three real zeros: Two of the zeros are irrational; only one is rational (- 3/4). The Rational Zero Test makes no claims beyond those for rational zeros. It simply states that if any rational zeros exist, they will appear on the computed list. Indeed, - 3/4 was on the computed list!

4. Table of Contents Polynomials: The Rational Zero Test Slide 4 Try: Make a list of possible rational zeros of the polynomial, P(x) = 5x 3 – 14x 2 + 3x + 4. The possible rational zeros are:

5. Table of Contents Polynomials: The Rational Zero Test