The polynomials $X^k(X-1)^{n-k}$ form a basis of $\mathbb{R_n}[X]$

Question

How to prove that the family of polynomials $X^k(X-1)^{n-k}$ for $k=0,\dots n$ is a basis of $\mathbb{R_n}[X]$. My plan is to prove those polynomials are linearly independent, that is if $$\sum_{k=0}^n\alpha_k X^k(X-1)^{n-k}=0$$ by derivating the above formula, but I noticed that all these polynomials have the same dergee $n$.

Answer 1

Hint. For each $0 \leq m \leq n$, we have

$$X^m = X^m(X - (X-1))^{n-m} = \sum_{k=0}^{n-m} \binom{n-m}{k} (-1)^k X^{n-k}(X-1)^k. $$