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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/.
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.
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 ...
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