Quadratic Polynomials

$1$ point possible $($graded$)$

Recall a degree $n$ polynomial in $x_1,x_2,$dots,$x_k$ is a linear combination of monomials in $x_1,x_2,$dots,$x_k$ where

the highest of degrees of the monomials with non$-$zero coefficients is $n.$

$($Recall monomials in $x_1,x_2,$dots,$x_k$ are unordered words using $x_1,x_2,$dots,$x_k$ as the letters, and the

degree of a monomial is its length $($or power$),$ e$.$g$.$ the monomial $x_1{x}_2{x}_3^2$:$=x_1{x}_2{x}_3{x}_3$ has degree $4.)$

Examples:

A degree $2,$ also known as quadratic, polynomial in the $1$ variable $x$ is of the form

$a{x}^2+bx+c$

for some numbers $a,b,c.$ The polynomial is determined by the $3$ coefficients $a,b,c,$ and different

choices of $(a,b,c)$ result in different polynomials.

In linear algebraic terms, the space of degree $2$ polynomials in $1$ variable is of dimension $3$ since it

consists of all linear combinations of $3$ linearly independent vectors $x^2,x,$ and $1.$

A degree $2$ polynomial in $2$ variables $x_1,x_2$ is of the form

$a{x}_1^2+b{x}_2^2+c{x}_1{x}_2+d{x}_1+e{x}_2+f$

for some numbers $a,b,c,d,e,f.$ Different choices of $(a,b,c,d,e,f)$ result in different polynomials.

In linear algebraic terms, the space of degree $2$ polynomials in $2$ variables is of dimension $6$ since it

consists of all linear combinations of $6$ linearly independent vectors $x_1^2,x_2^2,x_1{x}_2,x_1,x_2,$ and $1$