College Algebra Exam Review 37

# College Algebra Exam Review 37 - 1.8 POLYNOMIALS(d 47...

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: 1.8. POLYNOMIALS (d) 47 Multiplication in KŒx is commutative and associative; that is, for all f; g; h 2 KŒx, fg D gf; and (e) f .gh/ D .fg/h: 1 is an identity for multiplication; that is, for all f 2 KŒx, 1f D f: (f) The distributive law holds: For all f; g; h 2 KŒx, f .g C h/ D fg C f h: P Deﬁnition 1.8.3. The degree of a polynomial k ak x k is the largest k such that ak ¤ 0. (The degree of a constant polynomial c is zero, unless c D 0. By convention, the degree of the constant polynomial 0 is 1.) The degree of p 2 KŒx is denoted deg.p/. P If p D j aj x j is a nonzero polynomial of degree k , the leading coefﬁcient of p is ak and the leading term of p is ak x k . A polynomial is said to be monic if its leading coefﬁcient is 1. p3 2x is 7; the Example 1.8.4. The degree of p D . =2/x 7 C ix 4 leading coefﬁcient is =2; .2= /p is a monic polynomial. Proposition 1.8.5. Let f; g 2 KŒx. (a) deg.fg/ D deg.f / C deg.g/; in particular, if f and g are both nonzero, then fg ¤ 0. (b) deg.f C g/ Ä maxfdeg.f /; deg.g/g. Proof. Exercise 1.8.3. I We say that a polynomial f divides a polynomial g (or that g is divisible by f ) if there is a polynomial q such that f q D g . We write f jg for “f divides g .” The goal of this section is to show that KŒx has a theory of divisibility, or factorization, that exactly parallels the theory of divisibility for the integers, which was presented in Section 1.6. In fact, all of the results of this ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online