How to factor a variable - Algebra 1
Card 0 of 18
If 
, and 
and 
are distinct positive integers, what is the smallest possible value of 
?
If
Consider the possible values for (x, y):
(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.
Consider the possible values for (x, y):
(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.
Compare your answer with the correct one above
Solve for 
.
Solve for
Multiply both sides by 3:
Distribute:
Subtract 
from both sides:
Add the 
terms together, and subtract 
from both sides:
Divide both sides by 
:
Simplify:
Multiply both sides by 3:
Distribute:
Subtract
Add the
Divide both sides by
Simplify:
Compare your answer with the correct one above
Factor the following expression:
Factor the following expression:
Here you have an expression with three variables. To factor, you will need to pull out the greatest common factor that each term has in common.
Only the last two terms have 
so it will not be factored out. Each term has at least 
and 
so both of those can be factored out, outside of the parentheses. You'll fill in each term inside the parentheses with what the greatest common factor needs to be multiplied by to get the original term from the original polynomial:
Here you have an expression with three variables. To factor, you will need to pull out the greatest common factor that each term has in common.
Only the last two terms have
Compare your answer with the correct one above
Factor the following polynomial: 
.
Factor the following polynomial:
Because the 
term doesn’t have a coefficient, you want to begin by looking at the 
term (
) of the polynomial: 
. Find the factors of 
that when added together equal the second coefficient (the 
term) of the polynomial.
There are only four factors of 
: 
, and only two of those factors, 
, can be manipulated to equal 
when added together and manipulated to equal 
when multiplied together: 
(i.e.,
).
Because the
There are only four factors of
Compare your answer with the correct one above
Factor the following polynomial: 
.
Factor the following polynomial:
Because the 
term doesn’t have a coefficient, you want to begin by looking at the 
term (
) of the polynomial: 
.
Find the factors of 
that when added together equal the second coefficient (the 
term) of the polynomial: 
.
There are seven factors of 
: 
, and only two of those factors, 
, can be manipulated to equal 
when added together and manipulated to equal 
when multiplied together: 
Because the
Find the factors of
There are seven factors of
Compare your answer with the correct one above
Solve for 
when 
: 
Solve for
First, factor the numerator: 
.
Now your expression looks like
Second, cancel the "like" terms - 
- which leaves us with 
.
Third, solve for 
, which leaves you with 
.
First, factor the numerator:
Now your expression looks like
Second, cancel the "like" terms -
Third, solve for
Compare your answer with the correct one above
Factor the following polynomial: 
.
Factor the following polynomial:
Because the term has a coefficient, you begin by multiplying the 
and the 
terms () together: 
.
Find the factors of 
that when added together equal the second coefficient (the 
term) of the polynomial: 
.
There are four factors of 
: 
, and only two of those factors, 
, can be manipulated to equal 
when added together and manipulated to equal 
when multiplied together: 
Because the
Find the factors of
There are four factors of
Compare your answer with the correct one above
Solve for 
, when 
:
Solve for
First, factor the numerator, which should be 
. Now the left side of your equation looks like
Second, cancel the "like" terms - 
- which leaves us with 
.
Third, solve for 
by setting the left-over factor equal to 0, which leaves you with
First, factor the numerator, which should be
Second, cancel the "like" terms -
Third, solve for
Compare your answer with the correct one above
Two consecutive odd numbers have a product of 195. What is the sum of the two numbers?
Two consecutive odd numbers have a product of 195. What is the sum of the two numbers?
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:
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
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
Now that you have solved for
Compare your answer with the correct one above
Factor the expression:
Factor the expression:
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: 
.
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
Compare your answer with the correct one above
Factor the expression:
Factor the expression:
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: 
.
To find the greatest common factor, we must break each term into its prime factors:
The terms have
Pull this out of the expression to find the answer:
Compare your answer with the correct one above
Factor:
Factor:
The common factor here is 
. Pull this out of both terms to simplify:
The common factor here is
Compare your answer with the correct one above
Factor the following polynomical expression completely, using the "factor-by-grouping" method.
Factor the following polynomical expression completely, using the "factor-by-grouping" method.
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:
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
The GCF of the second group is
We can rewrite the original expression,
as,
The common factor for BOTH of these terms is
Dividing both sides by
Compare your answer with the correct one above
Factor the polynomial expression completely, using the "factor-by-grouping" method.
Factor the polynomial expression completely, using the "factor-by-grouping" method.
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:
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
(Remember, when dividing by a negative, the original number changes its sign!)
The GCF of the second group is
We can rewrite the original expression,
as,
The GCF of each of these terms is...
...so, the expression, when factored, is:
Compare your answer with the correct one above
Factor the following polynomial expression completely, using the "factor-by-grouping" method.
Factor the following polynomial expression completely, using the "factor-by-grouping" method.
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:
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
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
Factoring the second group by its GCF gives us:
We can rewrite the original expression:
which is the same as:
Compare your answer with the correct one above
Factor the following polynomial expression completely, using the "factor-by-grouping" method.
Factor the following polynomial expression completely, using the "factor-by-grouping" method.
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:
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
The GCF of the second group is
We can rewrite the original expression,
as,
We can factor this as:
Compare your answer with the correct one above
Factor: 
Factor:
For each term in this expression, we will notice that each shares a variable of 
. This can be pulled out as a common factor.
There are no more common factors, and this is the reduced form.
The answer is: 
For each term in this expression, we will notice that each shares a variable of
There are no more common factors, and this is the reduced form.
The answer is:
Compare your answer with the correct one above
Factor: 
Factor:
For each term in this expression, we will notice that each shares a variable of . This can be pulled out as a common factor.
There are no more common factors, and this is the reduced form.
The answer is: 
For each term in this expression, we will notice that each shares a variable of
There are no more common factors, and this is the reduced form.
The answer is:
Compare your answer with the correct one above