You can
put this solution on YOUR website!I'll do the first three to get you started
# 1
Start with the given expression
Factor out the GCF
Now let's focus on the inner expression
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Looking at
we can see that the first term is
and the last term is
where the coefficients are 2 and 3 respectively.
Now multiply the first coefficient 2 and the last coefficient 3 to get 6. Now what two numbers multiply to 6 and add to the middle coefficient 1? Let's list all of the factors of 6:
Factors of 6:
1,2,3,6
-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 6
1*6
2*3
(-1)*(-6)
(-2)*(-3)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1
| First Number | Second Number | Sum | | 1 | 6 | 1+6=7 |
| 2 | 3 | 2+3=5 |
| -1 | -6 | -1+(-6)=-7 |
| -2 | -3 | -2+(-3)=-5 |
None of these pairs of factors add to 1. So the expression
cannot be factored
So
just remains as
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Answer:
So
factors to
# 2
Looking at
we can see that the first term is
and the last term is
where the coefficients are 1 and 18 respectively.
Now multiply the first coefficient 1 and the last coefficient 18 to get 18. Now what two numbers multiply to 18 and add to the middle coefficient 11? Let's list all of the factors of 18:
Factors of 18:
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 18
1*18
2*9
3*6
(-1)*(-18)
(-2)*(-9)
(-3)*(-6)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11
| First Number | Second Number | Sum | | 1 | 18 | 1+18=19 |
| 2 | 9 | 2+9=11 |
| 3 | 6 | 3+6=9 |
| -1 | -18 | -1+(-18)=-19 |
| -2 | -9 | -2+(-9)=-11 |
| -3 | -6 | -3+(-6)=-9 |
From this list we can see that 2 and 9 add up to 11 and multiply to 18
Now looking at the expression
, replace
with
(notice
adds up to
. So it is equivalent to
)
Now let's factor
by grouping:
Group like terms
Factor out the GCF of
out of the first group. Factor out the GCF of
out of the second group
Since we have a common term of
, we can combine like terms
So
factors to
So this also means that
factors to
(since
is equivalent to
)
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Answer:
So
factors to
# 3
Looking at
we can see that the first term is
and the last term is
where the coefficients are 1 and -15 respectively.
Now multiply the first coefficient 1 and the last coefficient -15 to get -15. Now what two numbers multiply to -15 and add to the middle coefficient -14? Let's list all of the factors of -15:
Factors of -15:
1,3,5,15
-1,-3,-5,-15 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -15
(1)*(-15)
(3)*(-5)
(-1)*(15)
(-3)*(5)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14
| First Number | Second Number | Sum | | 1 | -15 | 1+(-15)=-14 |
| 3 | -5 | 3+(-5)=-2 |
| -1 | 15 | -1+15=14 |
| -3 | 5 | -3+5=2 |
From this list we can see that 1 and -15 add up to -14 and multiply to -15
Now looking at the expression
, replace
with
(notice
adds up to
. So it is equivalent to
)
Now let's factor
by grouping:
Group like terms
Factor out the GCF of
out of the first group. Factor out the GCF of
out of the second group
Since we have a common term of
, we can combine like terms
So
factors to
So this also means that
factors to
(since
is equivalent to
)
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Answer:
So
factors to
(