Processing math: 0%
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.
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.

Example.

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.