Polynomial Functions

An expression $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ (with $a_i\in R$ and $n$ a non-negative integer) is a polynomial in $x$. When $a_n\ne0$, the polynomial has degree $n$; $a_n$ is its leading coefficient and $a_0$ its constant term. A polynomial function $P(x)=a_nx^n+\cdots+a_0$ is defined on all of $R$; this chapter treats 'polynomial' and 'polynomial function' as the same thing.

Naming by degree: degree $1$ = linear, $2$ = quadratic, $3$ = cubic, $4$ = quartic, $5$ = quintic; any nonzero constant is a degree-$0$ polynomial.

Equality. Two polynomials $f(x)=a_nx^n+\cdots+a_0$ ($a_n\ne0$) and $g(x)=b_mx^m+\cdots+b_0$ ($b_m\ne0$) are equal (as functions, for every $x\in R$) exactly when $n=m$ and $a_k=b_k$ for every $k$ -- matching degree and matching every coefficient. This single fact is what makes the method of undetermined coefficients (§2.6.3) work.

Sum and product. Adding two polynomials adds corresponding coefficients, and $\deg(P+Q)\le\max(\deg P,\deg Q)$; multiplying multiplies every term of one by every term of the other, and $\deg(PQ)=\deg P+\deg Q$. …