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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 1 and -21 respectively.
Now multiply the first coefficient 1 and the last coefficient -21 to get -21. Now what two numbers multiply to -21 and add to the middle coefficient 4? Let's list all of the factors of -21:
Factors of -21:
1,3,7,21
-1,-3,-7,-21 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -21
(1)*(-21)
(3)*(-7)
(-1)*(21)
(-3)*(7)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 4
| First Number | Second Number | Sum | | 1 | -21 | 1+(-21)=-20 |
| 3 | -7 | 3+(-7)=-4 |
| -1 | 21 | -1+21=20 |
| -3 | 7 | -3+7=4 |
From this list we can see that -3 and 7 add up to 4 and multiply to -21
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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So our expression goes from
and factors further to
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Answer:
So
completely factors to
(