Stitz-Zeager_College_Algebra_e-book

# Theorem 39 rational zeros theorem suppose f x an xn

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online