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 









------------------------------------------------------------









     Answer:



So  factors to 







If you need more help, email me at jim_thompson5910@hotmail.com





Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you





Jim 

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