A functional property of homogeneous polynomials

Let $p(x_1,\ldots,x_n)$ be a homogeneous polynomial of degree $k$ in $R[x_1,\ldots,x_n]$. Prove that for all $\lambda \in R$, we have $p(\lambda x_1, \ldots, \lambda x_n) = \lambda^k p(x_1,\ldots,x_n)$.

Recall that a polynomial is called homogeneous of degree $k$ if it can be written in the form $p(x_1,\ldots,x_n) = \sum_{i=1}^m r_i \prod_{j=1}^n x_j^{k_{i,j}}$, where for each $i$, we have $\sum_{j=1}^n k_{i,j} = k$.

This being the case, we have $p(\lambda x_1, \ldots, \lambda x_n) = \sum_{i=1}^m r_i \prod_{j=1}^n (\lambda x_j)^{k_{i,j}}$ $= \sum_{i=1}^m r_i \prod \lambda^{k_{i,j}} x_j^{k_{i,j}}$ $= \sum_{i=1}^m r_i \lambda^{\sum k_{i,j}} \prod_{j=i}^n x_j^{k_{i,j}}$ $= \sum_{i=1}^m r_i \lambda^k \prod_{j=1}^n x_j^{k_{i,j}}$ $= \lambda_k \sum_{i=1}^m r_i \prod_{j=1}^n x_j^{k_{i,j}}$ $= \lambda^k p(x_1, \ldots, x_n)$.