Department of Mathematics

Polynomials and Factoring

Worked Examples

Adding, Subtracting, and Multiplying Polynomials

To add and subtract polynomials, we "collect like terms", i.e. combine (add or subtract) the coefficients of the terms which have the same variables and exponents.


Simplify $3xy+4x^2-5xy+8x^2$.
$3xy+4x^2-5xy+8x^2$$=3xy-5xy+4x^2+8x^2$ (Details)Reorganize the order of the terms so those with the same variable expression are next to each other.
$=(3-5)xy+(4+8)x^2$ (Details)Combine the coefficients of “like terms”

To multiply two polynomials, we need to take the product of one term in the first polynomial with a term in the second polynomial, repeat this for all such pairs, then add them up.


Expand $(x+2)(x^2+5x-1)$. $$\begin{align*} (x+2)(x^2+5x-1)=& x(x^2)+x(5x)+x(-1)+2(x^2)+2(5x)+2(-1) \\ =&x^3+5x^2-x+2x^2+10x-2 \\ =&x^3+(5+2)x^2+(10-1)x-2 \\ =&x^3+7x^2+9x-2 \end{align*}$$

Note: Multiplying two terms together involves combining their coefficients and combining their variables. $$\textrm{E.g. }\;\; (2xy^3)(7x^5y^2)=2(7)x(x^5)y^3y^2=14x^6y^5$$ We can also divide polynomials, this is the topic of the next section.