Chapter 1.3, Problem 49E

To determine

(a)

To prove:

$\delta \left(x^{k+1}\right)=\left(k+1\right)x^k+k{x}^{k-1}+\left(k-1\right)x^{k-2}+\mathrm{....}$.

To determine

(b)

To prove

There exists a polynomial with degree $\left(k+1\right)$ and leading coefficient $\frac{1}{\left(k+1\right)}$ :

$P\left(x\right)=\frac{1}{\left(k+1\right)}{x}^{k+1}+\frac{1}{k}{x}^k+\mathrm{...}$

Such that $\delta P={\left(x+1\right)}^k\text{ and }P\left(0\right)=0$