# Solving single-variable polynomials

## Single-variable polynomials

### Introduction

A single-variable polynomial is an equation of the form:

$$\sum^n_{i=0} a_i x^i=0$$

For example:

• $$x=1$$

• $$x^2=4$$

• $$x^2-3x+2=0$$

### Degrees

The degree of a polynomial is the highest-order term.

For example $$x^3+x=0$$ has degree $$3$$.

### Roots of single-variable polynomials

A solution to a polynomial is a root.

For example $$1$$ and $$2$$ are roots of $$x^2-3x+2=0$$

Quadratic polynomials are of the form $$ax^2+bx+c=0$$.

$$x=\dfrac{-b\pm \sqrt {b^2-4ac}}{2a}$$

### Proof

We can get the two solutions to a quadratic equation from the following manipulation.

$$ax^2+bx+c=0$$

$$a[x^2+\dfrac{b}{a}x]=-c$$

$$a[(x+\dfrac{b}{2a})^2-\dfrac{b^2}{4a^2}]=-c$$

$$a[(x+\dfrac{b}{2a})^2]=\dfrac{b^2}{4a}-c$$

$$(x+\dfrac{b}{2a})^2=\dfrac{b^2-4ac}{4a^2}$$

$$x+\dfrac{b}{2a}=\pm \sqrt {\dfrac{b^2-4ac}{4a^2}}$$

$$x=\dfrac{-b\pm \sqrt {b^2-4ac}}{2a}$$