1.7 Polynomials and their degrees

Degree 0: nonzero real numbers give constant polynomials $c$;

Degree 1: linear polynomials have the form $aX+b$;

Degree 2: Quadratic polynomials have the form $aX^{2}+bX+c$;

Degree 3: Cubic polynomials have the form $aX^{3}+bX^{2}+cX+d$

Degree 4: Quartic polynomials have the form $aX^{4}+bX^{3}+cX^{2}+dX+e$.

Note how many times the graphs can cross the $x$ axis, and how many maxima and minima they have.

Evaluating polynomials

Given $b\in{\mathbb{R}}$, we can evaluate $f(X)$ at $b$ by working out the terms, so

We substitute $b$ for the indeterminate $X$, so replace $X^{2}$ by $b^{2}$, and so on. Formally, there is a map $X\mapsto b$ and $f(X)\mapsto f(b).$