# Polynomial of degree 2: what happens when variable triples?

Let p(x,y,z) be a homogeneous polynomial of degree 2:

if p(2,3,4) = 10, what is p(6,9,12)?

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By the definition of homogeneous of degree $2$, we know $p(\lambda x,\lambda y,\lambda z)=\lambda^2 p(x,y,z)$ (this is easy to verify just by looking at the equation). So if we let $\lambda=3$, $x=2, y=3, z=4$, then we obtain $$p(6,9,12)=p(3\cdot2,3\cdot 3, 3\cdot 4)=3^2p(2,3,4)=9\cdot 10=90$$
Hint: if each variable is multiplied by $3$ and each term is a product of two ...
@user1561559: Do you know that $p(\lambda x,\lambda y,\lambda z)=\lambda^2 p(x,y,z)$? – wj32 Nov 16 '12 at 5:36