Polynomial Equations

Definition: Polynomial Equation

A polynomial equation over a field $F$ is an equation of the form

$$P(x_1,\cdots ,x_n)=0_F,$$

where $P(x_1,\cdots ,x_n)$ is a nonzero polynomial.

Definition: Solution of a Polynomial Equation

A solution to the polynomial equation $P(x_1,\cdots ,x_n)=0_F$ is an $n$-Tuples $S=({\lambda }_1,\cdots ,{\lambda }_n)$ of elements from $F$ such that the value of $P$ at $S$ is $0_F$.

Polynomial Equation in a Single Variable

A polynomial equation in a single variable has the form

$$a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0$$

Definition: Quadratic Equation

A quadratic equation is a polynomial equation $P=0$, where $P$ is a quadratic polynomial.

Quadratic Equation in a Single Variable

A quadratic equation in a single variable has the form

$$a{x}^2+bx+c=0$$