Homogeneous polynomial proof

Question

I have the homogeneous polynomial P=$\Sigma_0^k a_jx^jy^{k-j}$ where the vector x=(x,y)

I am trying to show

$x$.(($\nabla P)(x,y)) = x\frac{\partial P}{\partial x} + y \frac{\partial P}{\partial y}$ = $\lambda P(x,y)$ for a natural number $\lambda$ but have no idea where to start

Any help will be appreciated

Thank you

Answer 1

Hint: Fix $(x,y)$ and for $t\in \mathbb R$ define $ g(t) = P(t(x,y)).$ There are two ways to compute $g'(1).$