Let $C$ be an affine curve defined by a polynomial $p(x,y)$ of degree $d$. Show that if $(a,b)$ is a point of multiplicity $d$ in $C$ then $p(x,y)$ is a product of linear factors, so $C$ is union of d lines through $(a,b)$.
My attempt:
If $p(x,y)$ is homogeneous then by definition of tangent lines as in "Complex Algebraic Curves " by Kirwan we are done. But what if the curve is not homogeneous? Any help is appreciated!
Up to composing $p$ with a translation we may assume that $(a,b) = (0,0)$. Then take the expansion in homogeneous terms $$ p(x,y) = p_0(x,y) + p_1(x,y) + \dots + p_d(x,y) $$ where each $p_j(x,y)$ is homogeneous of degree $j$. By the definition of algebraic multiplicity, $p_0 = p_1 = \dots = p_{d-1}= 0$ hence $$ p(x,y) = p_d(x,y) $$ is homogeneous and the conclusion follows as you observed in the post.