Polynomials in Two Variables

Polynomials $P(x,y)$ add a layer of complexity, often requiring techniques like homogenization or treating one variable as a constant to solve for the other.

Definition and Structure

A polynomial in two variables is a sum of terms $c{x}^i{y}^j$.

The degree of a term $x^i{y}^j$ is $i+j$. The degree of $P$ is the maximum degree of its terms.

Homogeneous Polynomials

A polynomial $P(x,y)$ is homogeneous of degree $n$ if every term has degree $n$.

• Property: $P(tx,ty)=t^{n}P(x,y)$.
• Factoring: Homogeneous polynomials can often be factored into linear forms over $\mathbb{C}$:

Symmetry

A polynomial is symmetric if $P(x,y)=P(y,x)$.

• Any symmetric polynomial can be expressed in terms of the elementary symmetric polynomials:

Factorization Techniques

1. Treating One Variable as Constant

To factor $P(x,y)$, arrange it as a polynomial in $x$: Then find the discriminant ${\Delta }_x=B(y{)}^2-4A(y)C(y)$. If ${\Delta }_x$ is a perfect square, $P(x,y)$ factors.

Practice Problems