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Unformatted text preview: ll of the polynomials in Theorem 3.4 have special names. The polynomial p
is called the dividend; d is the divisor; q is the quotient; r is the remainder. If r(x) = 0 then
d is called a factor of p. The proof of Theorem 3.4 is usually relegated to a course in Abstract
Algebra,3 but we will use the result to establish two important facts which are the basis of the rest
of the chapter.
Theorem 3.5. The Remainder Theorem: Suppose p is a polynomial of degree at least 1 and
c is a real number. When p(x) is divided by x − c the remainder is p(c).
The proof of Theorem 3.5 is a direct consequence of Theorem 3.4. When a polynomial is divided
by x − c, the remainder is either 0 or has degree less than the degree of x − c. Since x − c is degree
1, this means the degree of the remainder must be 0, which means the remainder is a constant.
Hence, in either case, p(x) = (x − c) q (x) + r, where r, the remainder, is a real number, possibly
0. It follows that p(c) = (c − c) q (c) + r = 0 · q (c) + r = r, and s...
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