What if you had a polynomial expression like in which some of the terms shared a common factor but not all of them? How could you factor this expression? After completing this Concept, you'll be able to factor polynomials like this one by grouping.

### Watch This

CK-12 Foundation: 0913S Factoring By Grouping

### Guidance

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 next example illustrates how this process works.

#### Example A

*Factor* .

**Solution**

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 . Factor 2 from the first two terms and factor from the last two terms:

Now we notice that the binomial is common to both terms. We factor the common binomial and get:

#### Example B

*Factor* .

**Solution**

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

Now factor from both terms: .

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 , where .

A quadratic like this doesn’t factor as , so it’s not as simple as looking for two numbers that multiply to and add up to . Instead, we also have to take into account the coefficient in the first term.

To factor a quadratic polynomial where , we follow these steps:

- We find the product .
- We look for two numbers that multiply to and add up to .
- 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.

#### Example C

*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: , so the problem becomes:

*Step 4:* Factor an from the first two terms and a 2 from the last two terms:

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: , so the problem becomes:

*Step 4:* Factor by grouping: factor a from the first two terms and a -4 from the last two terms:

Now factor the common binomial :

Watch this video for help with the Examples above.

CK-12 Foundation: Factoring By Grouping

### Vocabulary

- It is possible to factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called
**factoring by grouping**.

### Guided Practice

*Factor by grouping.*

**Solution:**

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: , so the problem becomes:

*Step 4:* Factor by grouping: factor an from the first two terms and from the last two terms:

Now factor the common binomial :

### Practice

Factor by grouping.

Factor the following quadratic trinomials by grouping.