# Algebra 1 : Factoring Polynomials

## Example Questions

### Example Question #1 : Simplifying Expressions

Factor the expression:

Explanation:

To find the greatest common factor, we need to break each term into its prime factors:

Looking at which terms all three expressions have in common; thus, the GCF is . We then factor this out:

### Example Question #1 : How To Factor A Variable

Factor the expression:

Explanation:

To find the greatest common factor, we must break each term into its prime factors:

The terms have , and  in common; thus, the GCF is .

Pull this out of the expression to find the answer:

### Example Question #1 : How To Factor A Variable

If , and  and  are distinct positive integers, what is the smallest possible value of ?

Explanation:

Consider the possible values for (xy):

(1, 100)

(2, 50)

(4, 25)

(5, 20)

Note that (10, 10) is not possible since the two variables must be distinct. The sums of the above pairs, respectively, are:

1 + 100 = 101

2 + 50 = 52

4 + 25 = 29

5 + 20 = 25, which is the smallest sum and therefore the correct answer.

### Example Question #1 : How To Factor A Variable

Solve for .

Explanation:

Multiply both sides by 3:

Distribute:

Subtract  from both sides:

Add the  terms together, and subtract  from both sides:

Divide both sides by :

Simplify:

### Example Question #1 : How To Factor A Variable

Factor the following polynomical expression completely, using the "factor-by-grouping" method.

Explanation:

Let's split the four terms into two groups, and find the GCF of each group.

First group:

Second group:

The GCF of the first group is .  When we divide the first group's terms by , we get: .

The GCF of the second group is .  When we divide the second group's terms by , we get: .

We can rewrite the original expression,

as,

The common factor for BOTH of these terms is .

Dividing both sides by gives us:

### Example Question #1 : How To Factor A Variable

Factor the following polynomial expression completely, using the "factor-by-grouping" method.

Explanation:

Let's separate the four terms of the polynomial expression into two groups, and then find the GCF (greatest common factor) for each group.

First group:

Second group:

The GCF of the first group is ; it's the only factor both terms have in common.  Factoring the first group by its GCF gives us:

The second group is a bit tricky.  It looks like they have no factor in common.  But, each of the terms can be divided by !  So, the GCF is .

Factoring the second group by its GCF gives us:

We can rewrite the original expression:

is the same as:

,

which is the same as:

### Example Question #1 : How To Factor A Variable

Factor the following polynomial expression completely, using the "factor-by-grouping" method.

Explanation:

Separate the four terms into two groups, and then find the GCF of each group.

First group:

Second group:

The GCF of the first group is .  Factoring out from the terms in the first group gives us:

The GCF of the second group is .  Factoring out from the terms in the second group gives us:

We can rewrite the original expression,

as,

We can factor this as:

### Example Question #1 : How To Factor A Variable

Factor the polynomial expression completely, using the "factor-by-grouping" method.

Explanation:

Start by separating the four terms into two groups, and find the GCF (greatest common factor) of each group.

First group:

Second group:

The GCF of the first group is .  By factoring out from each term in the first group, we are left with:

(Remember, when dividing by a negative, the original number changes its sign!)

The GCF of the second group is .  By factoring out from each term in the second group, we get:

We can rewrite the original expression,

as,

The GCF of each of these terms is...

,

...so, the expression, when factored, is:

### Example Question #11 : How To Factor A Variable

Two consecutive odd numbers have a product of 195. What is the sum of the two numbers?

Explanation:

You can set the two numbers to equal variables, so that you can set up the algebra in this problem. The first odd number can be defined as  and the second odd number, since the two numbers are consecutive, will be .

This allows you to set up the following equation to include the given product of 195:

Next you can subtract 195 to the left and set the equation equal to 0, which allows you to solve for :

You can factor this quadratic equation by determining which factors of 195 add up to 2. Keep in mind they will need to have opposite signs to result in a product of negative 195:

Set each binomial equal to 0 and solve for . For the purpose of this problem, you'll only make use of the positive value for :

Now that you have solved for , you know the two consecutive odd numbers are 13 and 15. You solve for the answer by finding the sum of these two numbers:

Factor: