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Question 201026: factor the triominal
a^3 - 2a - 63a
Found 2 solutions by RAY100, jim_thompson5910:
Answer by RAY100(1637)   (Show Source):
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website! I'm going to assume you meant to say 
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 -63 respectively.
Now multiply the first coefficient 1 and the last coefficient -63 to get -63. Now what two numbers multiply to -63 and add to the middle coefficient -2? Let's list all of the factors of -63:
Factors of -63:
1,3,7,9,21,63
-1,-3,-7,-9,-21,-63 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -63
(1)*(-63)
(3)*(-21)
(7)*(-9)
(-1)*(63)
(-3)*(21)
(-7)*(9)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2
| First Number | Second Number | Sum | | 1 | -63 | 1+(-63)=-62 | | 3 | -21 | 3+(-21)=-18 | | 7 | -9 | 7+(-9)=-2 | | -1 | 63 | -1+63=62 | | -3 | 21 | -3+21=18 | | -7 | 9 | -7+9=2 |
From this list we can see that 7 and -9 add up to -2 and multiply to -63
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
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