Department of Mathematics

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. $p(a)=0$.


$x=1$ is a root of the polynomial $\; p(x)=x^5-5x^4+2x^3+x+1\; $ since: $$\begin{align*} p(1)&=(1)^5-5(1)^4+2(1)^3+1+1 \\ &=1-5+2+1+1 \\ &=0. \end{align*}$$

Example. Change the constants $a,b,c,d,e$ to see the graphs of different polynomials.