Question

Is the quadratic form x₁²+x₂²+2x₁x₃+4x₁x₃+3x₃² quadratic? Justify your answer

Answer 1

A quadratic form is a polynomial with terms all of degree two. 

Quadratic forms are homogeneous quadratic polynomials in $n$ variables.

In our case $n=3,$ and we have ternary quadratic form

$f(x_1, x_2, x_3)=1\cdot x_1^2+0\cdot x_1x_2+6\cdot x_1x_3+1\cdot x_2^2$

$$+0\cdot x_2x_3+1\cdot x_3^2$$

The quadratic form $f(x_1, x_2, x_3)$ is determined by the matrix

$$A=\begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & 0 \\ 3 & 0 & 1 \end{bmatrix}$$