Question 173291: Can you please explain the following in full depth:
3x^2+26xy+16y^2
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 3 and 16 respectively.
Now multiply the first coefficient 3 and the last coefficient 16 to get 48. Now what two numbers multiply to 48 and add to the middle coefficient 26? Let's list all of the factors of 48:
Factors of 48:
1,2,3,4,6,8,12,16,24,48
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 48
1*48
2*24
3*16
4*12
6*8
(-1)*(-48)
(-2)*(-24)
(-3)*(-16)
(-4)*(-12)
(-6)*(-8)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 26? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 26
| First Number | Second Number | Sum | | 1 | 48 | 1+48=49 | | 2 | 24 | 2+24=26 | | 3 | 16 | 3+16=19 | | 4 | 12 | 4+12=16 | | 6 | 8 | 6+8=14 | | -1 | -48 | -1+(-48)=-49 | | -2 | -24 | -2+(-24)=-26 | | -3 | -16 | -3+(-16)=-19 | | -4 | -12 | -4+(-12)=-16 | | -6 | -8 | -6+(-8)=-14 |
From this list we can see that 2 and 24 add up to 26 and multiply to 48
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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