# What is Factoring by Grouping?

Jun 28, 2018

#### Explanation:

When we have a polynomial of the form

$a {x}^{2} + b x + c$

We can factor this quadratic with splitting up the $b$ term into two terms. This allows us to factor the left side of the expression and the right side individually, and look for a common factor between them. This is factoring by grouping.

Let's take our polynomial

$a {x}^{2} + b x + c$ again. To factor by grouping, we can rewrite this expression as

$\textcolor{b l u e}{a} {x}^{2} + b x + \textcolor{red}{c} = \textcolor{b l u e}{a} {x}^{2} + \left(\textcolor{b l u e}{a} + \textcolor{red}{c}\right) x + \textcolor{red}{c}$

Notice that $\left(a + c\right) x$ is the same as our $b$ term. We can distribute the $x$ to both terms to get

$\textcolor{b l u e}{a {x}^{2} + a x} + \textcolor{red}{c x + c}$

This is the essence of factoring by grouping. We can look at our polynomial as two groups of two terms.

From the blue terms, we can factor out an $a x$, and from the red terms, a $c$. This leaves us with

$\textcolor{b l u e}{a x} \textcolor{p u r p \le}{\left(x + 1\right)} + \textcolor{red}{c} \textcolor{p u r p \le}{\left(x + 1\right)}$

Now, both terms have an $x + 1$ in common, so we can factor that out to get

$\left(\textcolor{b l u e}{a x} + \textcolor{red}{c}\right) \textcolor{p u r p \le}{\left(x + 1\right)}$

We will not always have an $x + 1$ term. For instance, take the following polynomial:

$3 {x}^{2} + 11 x + 6$

Let's rewrite this as

$\textcolor{t u r q u o i s e}{3 {x}^{2} + 9 x} + \textcolor{\mathmr{and} a n \ge}{2 x + 6}$

We can factor a $3 x$ out of the blue terms, and a $2$ out of the orange terms. We get

$\textcolor{t u r q u o i s e}{3 x} \left(x + 3\right) + \textcolor{\mathmr{and} a n \ge}{2} \left(x + 3\right)$

We can factor an $x + 3$ out to get

$\left(3 x + 2\right) \left(x + 3\right)$

The key point is that we can rewrite our $b$ term as the sum of two terms so we can factor twice. Next, we look for a common factor between our newly factored expression.

Factoring by grouping will not always work- at this point it may be a good idea to resort to the Quadratic Formula.

Hope this helps!