Abstract Algebra Theory and Applications - Computer Science ...

258 CHAPTER 15 POLYNOMIALSExample 1. Suppose thatp(x) = 3 + 0x + 0x 2 + 2x 3 + 0x 4andq(x) = 2 + 0x − x 2 + 0x 3 + 4x 4are polynomials in Z[x]. If the coefficient of some term in a polynomialis zero, then we usually just omit that term. In this case we would writep(x) = 3 + 2x 3 and q(x) = 2 − x 2 + 4x 4 . The sum of these two polynomialsisThe product,p(x) + q(x) = 5 − x 2 + 2x 3 + 4x 4 .p(x)q(x) = (3 + 2x 3 )(2 − x 2 + 4x 4 ) = 6 − 3x 2 + 4x 3 + 12x 4 − 2x 5 + 8x 7 ,can be calculated either by determining the c i ’s in the definition or by simplymultiplying polynomials in the same way as we have always done. Example 2. Letandp(x) = 3 + 3x 3q(x) = 4 + 4x 2 + 4x 4be polynomials in Z 12 [x]. The sum of p(x) and q(x) is 7 + 4x 2 + 3x 3 + 4x 4 .The product of the two polynomials is the zero polynomial. This exampletells us that R[x] cannot be an integral domain if R is not an integral domain.Theorem 15.1 Let R be a commutative ring with identity. Then R[x] is acommutative ring with identity.Proof. Our first task is to show that R[x] is an abelian group underpolynomial addition. The zero polynomial, f(x) = 0, is the additive identity.Given a polynomial p(x) = ∑ ni=0 a ix i , the inverse of p(x) is easily verified tobe −p(x) = ∑ ni=0 (−a i)x i = − ∑ ni=0 a ix i . Commutativity and associativityfollow immediately from the definition of polynomial addition and from thefact that addition in R is both commutative and associative.