SOLUTION: Can someone please help? factor the expression 81x^2-54xy+9y^2 into a product of binomials.

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Question 407310: Can someone please help?
factor the expression 81x^2-54xy+9y^2 into a product of binomials.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

81x%5E2-54xy%2B9y%5E2 Start with the given expression


9%289x%5E2-6xy%2By%5E2%29 Factor out the GCF 9


Now let's focus on the inner expression 9x%5E2-6xy%2By%5E2




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Looking at 9x%5E2-6xy%2By%5E2 we can see that the first term is 9x%5E2 and the last term is y%5E2 where the coefficients are 9 and 1 respectively.

Now multiply the first coefficient 9 and the last coefficient 1 to get 9. Now what two numbers multiply to 9 and add to the middle coefficient -6? Let's list all of the factors of 9:



Factors of 9:
1,3

-1,-3 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 9
1*9
3*3
(-1)*(-9)
(-3)*(-3)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6

First NumberSecond NumberSum
191+9=10
333+3=6
-1-9-1+(-9)=-10
-3-3-3+(-3)=-6



From this list we can see that -3 and -3 add up to -6 and multiply to 9


Now looking at the expression 9x%5E2-6xy%2By%5E2, replace -6xy with -3xy-3xy (notice -3xy-3xy combines back to -6xy. So it is equivalent to -6xy)

9x%5E2%2Bhighlight%28-3xy-3xy%29%2By%5E2


Now let's factor 9x%5E2-3xy-3xy%2By%5E2 by grouping:


%289x%5E2-3xy%29%2B%28-3xy%2By%5E2%29 Group like terms


3x%283x-y%29-y%283x-y%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of -y out of the second group


%283x-y%29%283x-y%29 Since we have a common term of 3x-y, we can combine like terms

So 9x%5E2-3xy-3xy%2By%5E2 factors to %283x-y%29%283x-y%29


So this also means that 9x%5E2-6xy%2By%5E2 factors to %283x-y%29%283x-y%29 (since 9x%5E2-6xy%2By%5E2 is equivalent to 9x%5E2-3xy-3xy%2By%5E2)


note: %283x-y%29%283x-y%29 is equivalent to %283x-y%29%5E2 since the term 3x-y occurs twice. So 9x%5E2-6xy%2By%5E2 also factors to %283x-y%29%5E2



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So our expression goes from 9%289x%5E2-6xy%2By%5E2%29 and factors further to 9%283x-y%29%5E2


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Answer:

So 81x%5E2-54xy%2B9y%5E2 factors to 9%283x-y%29%5E2

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Jim