# Factoring by Grouping

## Use associative and commutative properties with factoring

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Factoring by Grouping

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.

### 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.

1. We find the product .
2. We look for two numbers that multiply to and add up to .
3. We rewrite the middle term using the two numbers we just found.
4. 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.

### 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.

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