Factoring Polynomials

Let's start by factoring out the greatest common factor. Then, we will factor the resulting trinomial.

Factor Out the Greatest Common Factor

The greatest common factor is a common factor of all the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. In this case, the greatest common factor is $x.$ The result of factoring out the greatest common factor from the given expression is a quadratic expression with a leading coefficient of $1.$ $$\begin{array}{c}x\left(x^2+5x+6\right)\end{array}$$ Let's temporarily only focus on the expression in parentheses, and bring back the greatest common factor after factoring.

Factor the Quadratic Expression

To factor a quadratic expression with a leading coefficient of $1,$ we first need to identify the values of $b$ and $c.$ $$\begin{array}{rl}\text{General Expression:}& x^2+bx+c\\ \text{Our Expression:}& x^2+5x+6\end{array}$$ Next, we have to find a factor pair of $c$ $=$ $6$ whose sum is $b$ $=$ $5.$ Note that $6$ is a positive number, so for the product of the factors to be positive, they must have the same sign — both positive or both negative.

Factor Pair	Product of Factors	Sum of Factors
$1$ and $6$	$6$	$7$
$\text{-}1$ and $\text{-}6$	$6$	$\text{-}7$
$2$ and $3$	$6$	$5$
$\text{-}2$ and $\text{-}3$	$6$	$\text{-}5$

The factors whose product is $6$ and whose sum is $5$ are $2$ and $3.$ With this information, we can now factor the trinomial. $$\begin{array}{c}{x}^2+5x+6\iff (x+2)(x+3)\end{array}$$ Before we finish, remember that we factored out the greatest common factor from the original expression. Therefore, we need to include it again. $$\begin{array}{c}{x}^3+5x^2+6x\iff x(x+2)(x+3)\end{array}$$