If \(a\) and \(b\) are elements of a field \(F\) and \(a \neq 0\), show that the polynomials \(1\), \(ax + b\), \((ax + b)^2\), \((ax + b)^3\), \(\ldots\) form a basis of \(F[x]\).
Solution
The polynomial \((ax+b)^k\) has degree \(k\) and leading coefficient \(a^k\ne0\). Hence the family \[ 1,\ ax+b,\ (ax+b)^2,\ldots \] is linearly independent because its elements have distinct degrees.
To show that it spans \(F[x]\), note that \[ x=\frac{(ax+b)-b}{a}. \] Therefore every monomial \(x^n\) is a polynomial in \(ax+b\), namely \[ x^n=a^{-n}\big((ax+b)-b\big)^n. \] Thus every standard monomial belongs to the span of the displayed family. Since the monomials span \(F[x]\), so does this family. Hence it is a basis of \(F[x]\).