5-5 Theorems About Roots of Polynomial Equations

Objectives

To solve equations using the Rational Root Theorem

To use the Conjugate Root Theorem

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Factoring the polynomial P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 p open x close equals , eh sub n , x to the n , plus . eh sub n minus 1 end sub . x super n minus 1 end super . plus math axis ellipsis plus , eh sub 1 , x plus , eh sub 0  can be challenging, especially when both a n eh sub n  and a 0 eh sub 0  have many factors.

Essential Understanding The factors of the numbers a n  and  a 0  in  P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 eh sub n , and , eh sub 0 , in p open x close equals , eh sub n , x to the n , plus . eh sub n minus 1 end sub . x super n minus 1 end super . plus math axis ellipsis plus , eh sub 1 , x plus , eh sub 0  can help you factor P(x) and solve the equation P ( x ) = 0 . p open x close equals 0 .

One way to find a root of the polynomial equation P ( x ) = 0 p open x close equals 0  is to guess and check. This is inefficient unless there is a way to minimize the number of guesses, or possible roots. The Rational Root Theorem does just that.

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