College Algebra Exam Review 44

College Algebra Exam Review 44 - 54 1 ALGEBRAIC THEMES...

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Unformatted text preview: 54 1. ALGEBRAIC THEMES factor polynomials in QŒx by exhaustive search, as there are infinitely many polynomials of each degree. We finish this section with some elementary but important results relating roots of polynomials to divisibility. Proposition 1.8.22. Let p 2 KŒx and a 2 K . Then there is a polynomial q such that p.x/ D q.x/.x a/ C p.a/. Consequently, p.a/ D 0 if, and only if, x a divides p . Proof. Write p.x/ D q.x/.x a/ C r , where the remainder r is a constant. Substituting a for x gives p.a/ D r . I Definition 1.8.23. Say an element ˛ 2 K is a root of a polynomial p 2 KŒx if p.˛/ D 0. Say the multiplicity of the root ˛ is k if x ˛ appears exactly k times in the irreducible factorization of p . Corollary 1.8.24. A polynomial p 2 KŒx of degree n has at most n roots in K , counting with multiplicities. That is, the sum of multiplicities of all roots is at most n. Proof. If p D .x ˛1 /m1 .x ˛2 /m2 .x ˛k /mk q1 qs , where the qi are irreducible, then evidently m1 C m2 C C mk Ä deg.p/. I Exercises 1.8 Exercises 1.8.1 through 1.8.2 ask you to prove parts of Proposition 1.8.2. 1.8.1. Prove that addition in KŒx is commutative and associative, that 0 is an identity element for addition in KŒx, and that f C f D 0 for all f 2 KŒx. 1.8.2. Prove that multiplication in KŒx is commutative and associative, and that 1 is an identity element for multiplication. 1.8.3. Prove Proposition 1.8.5. 1.8.4. Prove Proposition 1.8.15. ...
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