How to factor a trinomial - Algebra 1
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For what value of 
allows one to factor a perfect square trinomial out of the following equation:
For what value of
Factor out the 7:
Take the 8 from the x-term, cut it in half to get 4, then square it to get 16. Make this 16 equal to C/7:
Solve for C:
Factor out the 7:
Take the 8 from the x-term, cut it in half to get 4, then square it to get 16. Make this 16 equal to C/7:
Solve for C:
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Factor the following trinomial: 
.
Factor the following trinomial:
To factor trinomials like this one, we need to do a reverse FOIL. In other words, we need to find two binomials that multiply together to yield 
.
Finding the "first" terms is relatively easy; they need to multiply together to give us 
, and since 
only has two factors, we know the terms must be 
and 
. We now have 
 
, and this is where it gets tricky.
The second terms must multiply together to give us 
, and they must also multiply with the first terms to give us a total result of 
. Many terms fit the first criterion. 
, 
, 
and 
all multiply to yield 
. But the only way to also get the "
" terms to sum to 
is to use 
. It's just like a puzzle!
To factor trinomials like this one, we need to do a reverse FOIL. In other words, we need to find two binomials that multiply together to yield
Finding the "first" terms is relatively easy; they need to multiply together to give us
The second terms must multiply together to give us
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Which of the following values of 
would make the trinomial 
prime?
Which of the following values of
For the trinomial 
to be factorable, we would have to be able to find two integers with product 36 and sum 
; that is, 
would have to be the sum of two integers whose product is 36.
Below are the five factor pairs of 36, with their sum listed next to them. 
must be one of those five sums to make the trinomial factorable.
1, 36: 37
2, 18: 20
3, 12: 15
4, 9: 13
6, 6: 12
Of the five choices, only 16 is not listed, so if 
, then the polynomial is prime.
For the trinomial
Below are the five factor pairs of 36, with their sum listed next to them.
1, 36: 37
2, 18: 20
3, 12: 15
4, 9: 13
6, 6: 12
Of the five choices, only 16 is not listed, so if
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Factor the trinomial.
Factor the trinomial.
Use the 
-method to split the middle term into the sum of two terms whose coefficients have sum 
and product 
. These two numbers can be found, using trial and error, to be 
and 
.
and 
Now we know that 
is equal to 
.
Factor by grouping.
Use the
Now we know that
Factor by grouping.
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Factor completely: 
Factor completely:
We are looking to factor this quadratic trinomial into two factors
, where the question marks are to be replaced by two integers whose product is 
and whose sum is 
.
We need to look at the factor pairs of 
in which the negative number has the greater absolute value, and see which one has sum 
:
None of these pairs have the desired sum, so the polynomial is prime.
We are looking to factor this quadratic trinomial into two factors
We need to look at the factor pairs of
None of these pairs have the desired sum, so the polynomial is prime.
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Factor completely: 
Factor completely:
We are looking to factor this quadratic trinomial into two factors
, where the question marks are to be replaced by two integers whose product is 
and whose sum is 
.
We need to look at the factor pairs of 
in which the negative number has the greater absolute value and the sum is 
:
None of these pairs have the desired sum, so the polynomial is prime.
We are looking to factor this quadratic trinomial into two factors
We need to look at the factor pairs of
None of these pairs have the desired sum, so the polynomial is prime.
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Factor completely: 
Factor completely:
First, we note that the coefficients have an LCD of 3, so we can factor that out:
We try to factor further by factoring quadratic trinomial 
. We are looking to factor it into two factors
, where the question marks are to be replaced by two integers whose product is 
and whose sum is 
.
We need to look at the factor pairs of 
in which the negative number has the greater absolute value, and see which one has sum 
:
None of these pairs have the desired sum, so 
is prime. 
is the complete factorization.
First, we note that the coefficients have an LCD of 3, so we can factor that out:
We try to factor further by factoring quadratic trinomial
We need to look at the factor pairs of
None of these pairs have the desired sum, so
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Factor completely: 
Factor completely:
Rewrite this as 
Use the 
-method by splitting the middle term into two terms, finding two integers whose sum is 1 and whose product is 
; these integers are 
, so rewrite this trinomial as follows:
Now, use grouping to factor this:
Rewrite this as
Use the
Now, use grouping to factor this:
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Factor the trinomial 
.
Factor the trinomial
We can factor this trinomial using the FOIL method backwards. This method allows us to immediately infer that our answer will be two binomials, one of which begins with 
and the other of which begins with 
. This is the only way the binomials will multiply to give us 
.
The next part, however, is slightly more difficult. The last part of the trinomial is 
, which could only happen through the multiplication of 1 and 2; since the 2 is negative, the binomials must also have opposite signs.
Finally, we look at the trinomial's middle term. For the final product to be 
, the 1 must be multiplied with the 
and be negative, and the 2 must be multiplied with the 
and be positive. This would give us 
, or the 
that we are looking for.
In other words, our answer must be
to properly multiply out to the trinomial given in this question.
We can factor this trinomial using the FOIL method backwards. This method allows us to immediately infer that our answer will be two binomials, one of which begins with
The next part, however, is slightly more difficult. The last part of the trinomial is
Finally, we look at the trinomial's middle term. For the final product to be
In other words, our answer must be
to properly multiply out to the trinomial given in this question.
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Factor this expression: 
Factor this expression:
When we factor, we have to remember to check the signs in the trinomial. In this case, we have minus 4 and minus 12. That automatically tells us the signs in the factors must be opposite, one plus and one minus.
Next, we ask ourselves what are the factors of 12? We get 2 & 6, 1 & 12, and 4 & 3.
Then, we ask ourselves, which of these, when subtracted in a given order, will give us -4? The answer is 6 and 2! So we place these in the parentheses with the 
's that we know go there so it looks like
Finally, we ask ourselves, what signs do we need to put in to get negative 4 and negative 12? We need a positive 
but a negative 
so we put the addition sign in with the 2 and the negative sign in with the 6!
When we factor, we have to remember to check the signs in the trinomial. In this case, we have minus 4 and minus 12. That automatically tells us the signs in the factors must be opposite, one plus and one minus.
Next, we ask ourselves what are the factors of 12? We get 2 & 6, 1 & 12, and 4 & 3.
Then, we ask ourselves, which of these, when subtracted in a given order, will give us -4? The answer is 6 and 2! So we place these in the parentheses with the
Finally, we ask ourselves, what signs do we need to put in to get negative 4 and negative 12? We need a positive
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Factor this trinomial: 
Factor this trinomial:
When we factor, we have to remember to check the signs in the trinomial. In this case, we have plus 9 and plus 18. That automatically tells us the signs in the factors must be the same, two "+".
Next, we ask ourselves what are the factors of 18? We get 2 & 9, 3 & 6, and 1 & 18.
Then, we ask ourselves, which of these when added in a given order, will give us 9? The answer is 6 and 3! So we place these in the parentheses with the 
's that we know go there so it looks like
Finally, we ask ourselves, what signs do we need to put in to get plus 9 and plus 18? We already answered that! Two "+" signs! So, the answer is:
When we factor, we have to remember to check the signs in the trinomial. In this case, we have plus 9 and plus 18. That automatically tells us the signs in the factors must be the same, two "+".
Next, we ask ourselves what are the factors of 18? We get 2 & 9, 3 & 6, and 1 & 18.
Then, we ask ourselves, which of these when added in a given order, will give us 9? The answer is 6 and 3! So we place these in the parentheses with the
Finally, we ask ourselves, what signs do we need to put in to get plus 9 and plus 18? We already answered that! Two "+" signs! So, the answer is:
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Factor this expression:
Factor this expression:
When we factor, we have to remember to check the signs in the trinomial. In this case, we have minus 7 and plus 12. That automatically tells us the signs in the factors must be the same, both "-" signs.
Next, we ask ourselves what are the factors of 12? We get 2 & 6, 1 & 12, and 4 & 3.
Then, we ask ourselves, which of these when added/subtracted in a given order, will give us 7? The answer is 4 and 3! So we place these in the parentheses with the 
's that we know go there so it looks like
When we factor, we have to remember to check the signs in the trinomial. In this case, we have minus 7 and plus 12. That automatically tells us the signs in the factors must be the same, both "-" signs.
Next, we ask ourselves what are the factors of 12? We get 2 & 6, 1 & 12, and 4 & 3.
Then, we ask ourselves, which of these when added/subtracted in a given order, will give us 7? The answer is 4 and 3! So we place these in the parentheses with the
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Factor this trinomial: 
Factor this trinomial:
When we factor, we have to remember to check the signs in the trinomial first. In this case, we have plus 11 and plus 14. That automatically tells us the signs in the factors must be the same, two "+" signs.
Next, we ask ourselves what are the factors of 14? We get 2 & 7 and 1 & 14.
Then, we need to account for the coefficient of 
. That 2 needs to be in front of one of the 
's in the binomials, so our two binomials will look like this so far:
Now we use those factors of 14 to fill in the blanks! We know it will be 2 and 7 because 14 plus 1 is more than 11! 2 plus 7 is less than 11. So, how will we get to 11? We know one of our constants will be multiplied by the 2 attached to the 
in the first binomial. Because multiplying 7 by 2 would get a number greater than 11, we know we put the 2 in the second binomial! That way the 
will multiply with the 2 and give us 
, which, added to 
(the result of multiplying the other variable and constant) will give us 
! So we know our answer will be:
When we factor, we have to remember to check the signs in the trinomial first. In this case, we have plus 11 and plus 14. That automatically tells us the signs in the factors must be the same, two "+" signs.
Next, we ask ourselves what are the factors of 14? We get 2 & 7 and 1 & 14.
Then, we need to account for the coefficient of
Now we use those factors of 14 to fill in the blanks! We know it will be 2 and 7 because 14 plus 1 is more than 11! 2 plus 7 is less than 11. So, how will we get to 11? We know one of our constants will be multiplied by the 2 attached to the
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Factor: 
Factor:
One way to factor a trinomial like this one is to put the terms of the polynomial into a box/grid:
Notice that there are 4 boxes but only 3 terms. To fix this, we find two numbers that add to -7 \[the middle coefficient\] and multiply to -30 \[the product of the first and last coefficients\]. By examining the factors of -30, we discover that these numbers must be -10 and 3.
Now we can put the terms into the box, by splitting the -7x into -10x and 3x:
To finish factoring, determine the greatest common factor of each of the rows and columns. For instance, 
and 
have a greatest common factor of 
, and 
and -15 have a greatest common factor of 3.
Our final answer just combines the factors on the top and side into binomials. In this case, 
.
One way to factor a trinomial like this one is to put the terms of the polynomial into a box/grid:
Notice that there are 4 boxes but only 3 terms. To fix this, we find two numbers that add to -7 \[the middle coefficient\] and multiply to -30 \[the product of the first and last coefficients\]. By examining the factors of -30, we discover that these numbers must be -10 and 3.
Now we can put the terms into the box, by splitting the -7x into -10x and 3x:
To finish factoring, determine the greatest common factor of each of the rows and columns. For instance,
Our final answer just combines the factors on the top and side into binomials. In this case,
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Factor: 
Factor:
One way to factor a trinomial like this one is to put the terms of the polynomial into a box/grid:
Notice that there are 4 boxes but only 3 terms. To fix this, we find two numbers that add to 22 \[the middle coefficient\] and multiply to -48 \[the product of the first and last coefficients\]. By examining the factors of -48, we discover that these numbers must be -2 and 24.
Now we can put the terms into the box, by splitting the 22x into -2x and 24x:
To finish factoring, determine the greatest common factor of each of the rows and columns. For instance, 
and 
have a greatest common factor of 
.
Our final answer just combines the factors on the top and side into binomials. In this case, 
.
One way to factor a trinomial like this one is to put the terms of the polynomial into a box/grid:
Notice that there are 4 boxes but only 3 terms. To fix this, we find two numbers that add to 22 \[the middle coefficient\] and multiply to -48 \[the product of the first and last coefficients\]. By examining the factors of -48, we discover that these numbers must be -2 and 24.
Now we can put the terms into the box, by splitting the 22x into -2x and 24x:
To finish factoring, determine the greatest common factor of each of the rows and columns. For instance,
Our final answer just combines the factors on the top and side into binomials. In this case,
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Factor the trinomial: 
Factor the trinomial:
The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of 
are 
:
Next, find the factors of 
. In this case, we could have 
or 
. Either combination can potentially produce 
, so the signage is important here.
Note that since the last term in the ordered trinomial is positive, both factors must have the same sign. Further, since the middle term in the ordered triniomial is negative, we know the signs must both be negative.
Therefore, we have two possibilities, 
or 
.
Let's solve for both, and check against the original triniomial.
Thus, our factors are 
and 
.
The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of
Next, find the factors of
Note that since the last term in the ordered trinomial is positive, both factors must have the same sign. Further, since the middle term in the ordered triniomial is negative, we know the signs must both be negative.
Therefore, we have two possibilities,
Let's solve for both, and check against the original triniomial.
Thus, our factors are
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Factor the trinomial: 
Factor the trinomial:
The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of 
are 
:
Next, find the factors of 
. In this case, we could have 
or 
. Since the factor on our leading term is 
, and no additive combination of 
and 
can create 
, we know that our factors must be 
.
Note that since the last term in the ordered trinomial is negative, the factors must have different signs.
Therefore, we have two possibilities, 
or 
.
Let's solve for both, and check against the original triniomial.
Thus, our factors are 
and 
.
The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of
Next, find the factors of
Note that since the last term in the ordered trinomial is negative, the factors must have different signs.
Therefore, we have two possibilities,
Let's solve for both, and check against the original triniomial.
Thus, our factors are
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Factor the trinomial: 
Factor the trinomial:
The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of are :
Next, find the factors of 
. In this case, we could have 
or 
. Since the factor on our leading term is , and no additive combination of 
and 
can create 
, we know that our factors must be 
.
Note that since the last term in the ordered trinomial is positive, the factors must have different signs. Since the middle term is also positive, the signs must both be positive.
Therefore, we have only one possibility, 
.
Let's solve, and check against the original triniomial.
Thus, our factors are 
and 
.
The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of
Next, find the factors of
Note that since the last term in the ordered trinomial is positive, the factors must have different signs. Since the middle term is also positive, the signs must both be positive.
Therefore, we have only one possibility,
Let's solve, and check against the original triniomial.
Thus, our factors are
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Factor this trinomial, then solve for 
: 
Factor this trinomial, then solve for
The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of are :
Next, find the factors of 
. In this case, we could have 
or 
. Since the factor on our leading term is , and no additive combination of 
and or of 
and 
can create our middle term of 
, we know that our factors must be 
.
Note that since the last term in the ordered trinomial is positive, the factors must have the same sign. Since the middle term is negative, both factors must have a negative sign.
Therefore, we have one possibility, 
.
Let's solve, and check against the original triniomial, before solving for 
.
Thus, our factors are and 
.
Now, let's solve for 
. Simply plug and play:
Therefore, 
and 
.
Note that, in algebra, we can represent this by showing 
. However, writing the word "and" is perfectly acceptable.
The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of
Next, find the factors of
Note that since the last term in the ordered trinomial is positive, the factors must have the same sign. Since the middle term is negative, both factors must have a negative sign.
Therefore, we have one possibility,
Let's solve, and check against the original triniomial, before solving for
Thus, our factors are
Now, let's solve for
Therefore,
Note that, in algebra, we can represent this by showing
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Factor the trinomial below,
Factor the trinomial below,
A factored trinomial is in the form 
, where 
the second term and 
the third term.
To factor a trinomial you first need to find the factors of the third term. In this case the third term is 
.
Factors of 
are: 
The factors you choose not only must multiply to equal the third term, they must also add together to equal the second term.
In this case they must equal 
.
To check your answer substitute the factors of 
into the binomials and use FOIL.
First terms: 
Outside terms: 
Inside terms: 
Last terms: 
Simplify from here by combining like terms
Using FOIL returned it to the original trinomial, therefore the answer is:
A factored trinomial is in the form
To factor a trinomial you first need to find the factors of the third term. In this case the third term is
Factors of
The factors you choose not only must multiply to equal the third term, they must also add together to equal the second term.
In this case they must equal
To check your answer substitute the factors of
First terms:
Outside terms:
Inside terms:
Last terms:
Simplify from here by combining like terms
Using FOIL returned it to the original trinomial, therefore the answer is:
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