Stitz-Zeager_College_Algebra_e-book

# You can blame the slow decline of civilization on him

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Unformatted text preview: o we get r = p(c), as required. There is one last ‘low hanging fruit’4 to collect - it is an immediate consequence of The Remainder Theorem. Theorem 3.6. The Factor Theorem: Suppose p is a nonzero polynomial. The real number c is a zero of p if and only if (x − c) is a factor of p(x). The proof of The Factor Theorem is a consequence of what we already know. If (x − c) is a factor of p(x), this means p(x) = (x − c) q (x) for some polynomial q . Hence, p(c) = (c − c) q (c) = 0, and so c is a zero of p. Conversely, if c is a zero of p, then p(c) = 0. In this case, The Remainder Theorem tells us the remainder when p(x) is divided by (x − c), namely p(c), is 0, which means (x − c) is a factor of p. What we have established is the fundamental connection between zeros of polynomials and factors of polynomials. Of the things The Factor Theorem tells us, the most pragmatic is that we had better ﬁnd a more eﬃcient way to divide polynomials by quantities of the form x − c. Fortunately, people like Ruﬃni and Horner have already blazed this trail. Let’s take a closer look at the long division we perf...
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## This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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