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PP Section 12.5 - Rational and Irrational Roots Objective...

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Rational and Irrational Roots
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Objective The equation x 2 – 3 = 0 has two irrational roots, while the equation x 2 – x – 6 = 0 has only rational roots. We want a tool that can tell us which rational numbers can be roots of a polynomial. Such a tool, together with other results, will allow us to also determine if a polynomial has irrational roots.
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Let f be a polynomial function of degree 1 or higher of the form f x a x a x a x a n n n n ( ) = + + + + - - 1 1 1 0 where each coefficient is an integer. If p / q , in lowest terms, is a rational zero of f , then p must be a factor of a 0 and q must be a factor of a n . The Rational Roots Theorem
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List the possible rational zeros of f x x x x ( ) = + - - 2 3 23 12 3 2 According the theorem, the numerators of a possible rational zeros will be factors of p = - 12 and the denominators will be factors of q = 2 Example Factors of p : ±1, ±2, ±3, ±4, ±6, ±12 Factors of q : ±1, ±2 12 , 6 , 4 , , 3 , 2 , 1, : zeros Possible 2 3 2 1 ± ± ± ± ± ± ± ±
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Example Find the rational zeros (if any) of 0 4 9 2 2 3 = - - - x x x
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