# 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.

### Example.

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” | ||

$=-2xy+12x^2$ |

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.

### Example.

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.