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4.4 Rational Root Theorem

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**Rational Root Theorem give direction in testing possible zeros.**

Let a0xn + a1xn-1 + …an-1x + an = 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 an and q is a factor of a0.

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**Ex 1 List all the possible rational roots then determine the rational roots.**

3x3 – 13x2 + 2x + 8 = 0

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Ex 2 Find ALL of the roots. x3 + 6x2 – 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) = 2x4 – x3 – 2x2 + 5x + 1

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