5.6 Find Rational Zeros.

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Presentation transcript:

5.6 Find Rational Zeros

The Rational Zero Theorem The polynomial function Has zeros: Notice the numerators of these zeros (-3, -5, and 7) are factors of the constant term –105. Notice the denominators of these zeros (2, 4, 8) are factors of the leading coefficient 64. The Rational Zero Theorem If Has integer coefficients, then every rational zero of has has the following form:

Since the coefficients are all integers, we can us the rational zero theorem. Find the rational zeros of Factors of 12: +1, +2, +3, +4, +6, +12 Factors of 1: + 1 Test x = 1 Test x = -1 1 2 -11 -12 1 2 -11 -12 1 -1 1 3 -8 -1 -1 12 1 3 -8 -20 1 1 -12 0 Since –1 is a zero of f, you can write the following: The zeros of f(x) are –1, 3, and –4.

Given: The lengthy list of possible rational zeros: With so many possibilities, it is worth our time to sketch the graph of the function. From the graph we can eliminate some choices and test others that seem reasonable. Reasonable choices are: Use these two given zeros and continue finding the other zeros.