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4.4 Rational Root Theorem

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Rational Root Theorem give direction in testing possible zeros. Let a 0 x n + a 1 x n-1 + …a n-1 x + a n = 0 represent a polynomial equation of degree n with integral coefficients. If a rational number p/q, where p and q have no common factors, is a root of the equation, then p is a factor of a n and q is a factor of a 0.

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Ex 1 List all the possible rational roots then determine the rational roots. 3x 3 – 13x 2 + 2x + 8 = 0

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Ex 2 Find ALL of the roots. x 3 + 6x 2 – 13x – 6 = 0

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Descartes Rule of Signs Used to determine the possible number of positive real zeros a polynomial has. P(x) is a polynomial in descending order. The # of positive real zeros is the same as the number of sign changes of the coefficients or is less than this by an even number. The # of negative real zeros is the same as the number of sign changes of the coefficients of P(- x), or less than by an even number. (Ignore zero coefficients.)

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Ex 3 find the number of possible positive and negative real zeros for then determine the rational zeros: P(x) = 2x 4 – x 3 – 2x 2 + 5x + 1

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