17: Polynomials

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Most people are fairly familiar with polynomials by the time they begin to study abstract algebra. When we examine polynomial expressions such as

\begin{align*} p(x) & = x^3 -3x +2\\ q(x) & = 3x^2 -6x +5\text{,} \end{align*}

we have a pretty good idea of what \(p(x) + q(x)\) and \(p(x) q(x)\) mean. We just add and multiply polynomials as functions; that is,

\begin{align*} (p +q)(x) & = p(x) + q(x)\\ & = ( x^3 - 3 x + 2 ) + ( 3 x^2 - 6 x + 5 )\\ & = x^3 + 3 x^2 - 9 x + 7 \end{align*}

and

\begin{align*} (p q)(x) & = p(x) q(x)\\ & = ( x^3 - 3 x + 2 ) ( 3 x^2 - 6 x + 5 )\\ & = 3 x^5 - 6 x^4 - 4 x^3 + 24 x^2 - 27 x + 10\text{.} \end{align*}

It is probably no surprise that polynomials form a ring. In this chapter we shall emphasize the algebraic structure of polynomials by studying polynomial rings. We can prove many results for polynomial rings that are similar to the theorems we proved for the integers. Analogs of prime numbers, the division algorithm, and the Euclidean algorithm exist for polynomials.