Express p (x) = x3 as a Taylor polynomial about a = - 1.
In calculus, you learn that a Taylor polynomial of degree n about a is a polynomial of the form
p(x) = a0 + a1(x - a) + a2(x - a)2 + · · · + an(x - a)n
where an ≠ 0. In other words, it is a polynomial that has been expanded in terms of powers of x - a instead of powers of x. Taylor polynomials are very useful for approximating functions that are "well behaved" near x = a.
The set B = {l, x - a, (x - a)2, . . . , (x - a)n} is a basis for Pn for any real number a. (Do you see a quick way to show this? Try using Theorem 6. 7.) This fact allows us to use the techniques of this section to rewrite a polynomial as a Taylor polynomial about a given a.