Question 407015: Please help,
factor the expression 81a^2-36ab+4b^2 into a product of binomials.
Found 2 solutions by ewatrrr, jim_thompson5910:
Answer by ewatrrr(24785)   (Show Source):
You can put this solution on YOUR website!
Hi
factoring into a product of binomials
81a^2-36ab+4b^2
(9a - 2b)(9a - 2b)
Note:SUM of the inner product(-18ab) and the outer product(-18ab) = -36ab
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 81 and 4 respectively.
Now multiply the first coefficient 81 and the last coefficient 4 to get 324. Now what two numbers multiply to 324 and add to the middle coefficient -36? Let's list all of the factors of 324:
Factors of 324:
1,2,3,4,6,9,12,18,27,36,54,81,108,162
-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 324
1*324
2*162
3*108
4*81
6*54
9*36
12*27
18*18
(-1)*(-324)
(-2)*(-162)
(-3)*(-108)
(-4)*(-81)
(-6)*(-54)
(-9)*(-36)
(-12)*(-27)
(-18)*(-18)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -36? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -36
| First Number | Second Number | Sum | | 1 | 324 | 1+324=325 | | 2 | 162 | 2+162=164 | | 3 | 108 | 3+108=111 | | 4 | 81 | 4+81=85 | | 6 | 54 | 6+54=60 | | 9 | 36 | 9+36=45 | | 12 | 27 | 12+27=39 | | 18 | 18 | 18+18=36 | | -1 | -324 | -1+(-324)=-325 | | -2 | -162 | -2+(-162)=-164 | | -3 | -108 | -3+(-108)=-111 | | -4 | -81 | -4+(-81)=-85 | | -6 | -54 | -6+(-54)=-60 | | -9 | -36 | -9+(-36)=-45 | | -12 | -27 | -12+(-27)=-39 | | -18 | -18 | -18+(-18)=-36 |
From this list we can see that -18 and -18 add up to -36 and multiply to 324
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 )
note: is equivalent to  since the term occurs twice. So also factors to
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Answer:
So factors to
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Jim
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