The Polar Form of a Quadratic Form

Suppose we have a quadratic form
$f(\mathbf{u}^T)=f(x_1,x_2,...,x_n)= \sum_{ij=1, \: i \leq=j}^n a_{ij}x_ix_j$
.
Let
$\mathbf{u}^T= (x_1,x_2,...,x_n), \: \mathbf{v}^T=(y_1, y_2,...,y_n)$

The polar form of this quadratic form is
$p(\mathbf{u}^T, \mathbf{v}^T)= \frac{1}{2}(f(\mathbf{u}^T+\mathbf{v}^T)-f(\mathbf{u}^T)-f(\mathbf{v}^T))$

Example: Let
$f(x_1,x_2)=x_1^2+3x_1x_2+x_2^2$

The
\begin{aligned} p((x_1,x_2),(y_1,y_2)) &= \frac{1}{2}((x_1+y_1)^2+3(x_1+y_1)(x_2+y_2)+(x_2+y_2)^2)-(x_1^2+3x_1x_2+x_2^2)-(y_1^2+3y_1x_2+y_2^2)\\ &= \frac{1}{2}(-6x_1x_2+2x_1y_1-3x_1y_2-3x_2y_1+2x_2y_2-6y_1y_2 \end{aligned}