### Factoring by Grouping

Sometimes, we can factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called **factor by grouping.**

The following examples illustrates how this process works.

*1. Factor*

There is no factor common to all the terms. However, the first two terms have a common factor of 2 and the last two terms have a common factor of

Now we notice that the binomial

*2. Factor*

We factor 3x from the first two terms and factor 4 from the last two terms:

Now factor

Now the polynomial is factored completely.

**Factor Quadratic Trinomials Where a ≠ 1**

Factoring by grouping is a very useful method for factoring quadratic trinomials of the form

A quadratic like this doesn’t factor as

To factor a quadratic polynomial where

- We find the product
ac . - We look for two numbers that multiply to
ac and add up tob . - We rewrite the middle term using the two numbers we just found.
- We factor the expression by grouping.

Let’s apply this method to the following examples.

#### Factoring by Grouping

Factor the following quadratic trinomials by grouping.

a)

b)

**Solution:**

Let’s follow the steps outlined above:

a)

*Step 1:*

*Step 2:* The number 12 can be written as a product of two numbers in any of these ways:

*Step 3:* Re-write the middle term:

*Step 4:* Factor an

Now factor the common binomial

To check if this is correct we multiply

The solution checks out.

b)

*Step 1:*

*Step 2:* The number 24 can be written as a product of two numbers in any of these ways:

*Step 3:* Re-write the middle term:

*Step 4:* Factor by grouping: factor a

Now factor the common binomial

### Example

#### Example 1

Factor

Let’s follow the steps outlined above:

*Step 1:*

*Step 2:* The number 5 can be written as a product of two numbers in any of these ways:

*Step 3:* Re-write the middle term:

*Step 4:* Factor by grouping: factor an

\begin{align*}x(5x-1)-1(5x-1)\end{align*}

Now factor the common binomial \begin{align*}(5x - 1)\end{align*}:

\begin{align*}(5x-1)(x-1) \qquad This \ is \ the \ answer.\end{align*}

### Review

Factor by grouping.

- \begin{align*}6x^2-9x+10x-15\end{align*}
- \begin{align*}5x^2-35x+x-7\end{align*}
- \begin{align*}9x^2-9x-x+1\end{align*}
- \begin{align*}4x^2+32x-5x-40\end{align*}
- \begin{align*}2a^2-6ab+3ab-9b^2\end{align*}
- \begin{align*}5x^2+15x-2xy-6y\end{align*}

Factor the following quadratic trinomials by grouping.

- \begin{align*}4x^2+25x-21\end{align*}
- \begin{align*}6x^2+7x+1\end{align*}
- \begin{align*}4x^2+8x-5\end{align*}
- \begin{align*}3x^2+16x+21\end{align*}
- \begin{align*}6x^2-2x-4\end{align*}
- \begin{align*}8x^2-14x-15\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.13.