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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 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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