Question 133000: Factor.
56 - 15w + w^2
I would appreciate any help you can give.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!  Start with the given equation
 Rearrange the terms
Looking at  we can see that the first term is  and the last term is  where the coefficients are 1 and 56 respectively.
Now multiply the first coefficient 1 and the last coefficient 56 to get 56. Now what two numbers multiply to 56 and add to the middle coefficient -15? Let's list all of the factors of 56:
Factors of 56:
1,2,4,7,8,14,28,56
-1,-2,-4,-7,-8,-14,-28,-56 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 56
1*56
2*28
4*14
7*8
(-1)*(-56)
(-2)*(-28)
(-4)*(-14)
(-7)*(-8)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -15
| First Number | Second Number | Sum | | 1 | 56 | 1+56=57 | | 2 | 28 | 2+28=30 | | 4 | 14 | 4+14=18 | | 7 | 8 | 7+8=15 | | -1 | -56 | -1+(-56)=-57 | | -2 | -28 | -2+(-28)=-30 | | -4 | -14 | -4+(-14)=-18 | | -7 | -8 | -7+(-8)=-15 |
From this list we can see that -7 and -8 add up to -15 and multiply to 56
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
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