# Polynomials and Factoring

## Worked Examples

## What is a Polynomial?

A**polynomial**is an expression involving numbers and variables raised to non-negative integer exponents. $$\textrm{E.g. }\; \; \; 7x+4x^5+9, \;\; xy^2-11, \;\; 2x, \;\; 32.$$

The

**terms**in a polynomial are the smaller expressions separated by "+" or "-". The terms are can be further broken down into

*coefficients*,

*variables*and

*exponents*. $$\textrm{E.g. } \; \; \; 7x+4x^5+9 \hspace{10 mm} \textrm{ Terms: }\; 7x, \; \; 4x^5, \;\; 9.$$ The term $4x^5$ has coefficient $4$, variable $x$ and exponent $5$.

The

**leading term**is the term with the highest exponent. The

**degree**of a polynomial is the exponent of the leading term. $$\textrm{E.g. } \; \; \; 7x+4x^5+9 \; \textrm{ has leading term }\; 4x^5 \; \textrm{ and degree } \; 5.$$

**Note:**Constant polynomials, e.g. $2$, have degree $0$ since $x^0=1$, so we have: $$2=2\cdot(1)=2x^0$$

A

**root**or a

**zero**of a polynomial in one variable, say $p(x)$, is a number $a$ such that substituting $x=a$ in the polynomial gives zero, i.e. $p(a)=0$.