The Rational Zero Theorem

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Presentation transcript:

The Rational Zero Theorem Let f(x) = anxn + an-1xn-1 + … + a1x + a0 represent a polynomial function with integer coefficients. If p/q in lowest terms is a rational zero of f, then p must be a factor of a0 (constant or last #) and q is a factor of an (leading coefficient).

a) State all possible rational zeros for g(x) = x3 + 2x2 - 3x + 5 Example 4 Factors of the constant Factors of the leading coefficient a) State all possible rational zeros for g(x) = x3 + 2x2 - 3x + 5 +5,+1 +1 Possible rational zeros: +5, +1 b) State all possible rational zeros for g(x) = 6x3 + 6x2 - 15x - 2 +2,+1 +6,+3,+2,+1 +2,+1 Possible rational zeros: +6,+3,+2,+1

Corollary If the coefficients of a polynomial function are integers such that an = 1 and a0 = 0, any rational zeros of the function must be factors of a0.

Try Check Your Progress #1A Answer

Example Find all the zeros of the function. From the corollary to the Fundamental Theorem of Algebra – we know that there are exactly 4 complex roots. Using Descartes Rule of signs, there are 4, 2, or 0 positive real roots and 0 negative real roots. Use the Rational Zeros Theorem: Test some of these and find that f(5) = 0 Depressed polynomial Factor Zeros are 5, 3/2, 2i, and –2i

Homework Assignment #39 p. 372 11-29 odd, 32-34, 44-45