SOLUTION: Factor completely. If the polynomial cannot be factored, enter PRIME 27x^2-36xy+12y^2 I'm clues on how to do this?help

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Question 319182: Factor completely. If the polynomial cannot be factored, enter PRIME
27x^2-36xy+12y^2
I'm clues on how to do this?help
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

27x%5E2-36xy%2B12y%5E2 Start with the given expression


3%289x%5E2-12xy%2B4y%5E2%29 Factor out the GCF 3


Now let's focus on the inner expression 9x%5E2-12xy%2B4y%5E2




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Looking at 9x%5E2-12xy%2B4y%5E2 we can see that the first term is 9x%5E2 and the last term is 4y%5E2 where the coefficients are 9 and 4 respectively.

Now multiply the first coefficient 9 and the last coefficient 4 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient -12? Let's list all of the factors of 36:



Factors of 36:
1,2,3,4,6,9,12,18

-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 36
1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)

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


Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12

First NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+6=12
-1-36-1+(-36)=-37
-2-18-2+(-18)=-20
-3-12-3+(-12)=-15
-4-9-4+(-9)=-13
-6-6-6+(-6)=-12



From this list we can see that -6 and -6 add up to -12 and multiply to 36


Now looking at the expression 9x%5E2-12xy%2B4y%5E2, replace -12xy with -6xy%2B-6xy (notice -6xy%2B-6xy adds up to -12xy. So it is equivalent to -12xy)

9x%5E2%2Bhighlight%28-6xy%2B-6xy%29%2B4y%5E2


Now let's factor 9x%5E2-6xy-6xy%2B4y%5E2 by grouping:


%289x%5E2-6xy%29%2B%28-6xy%2B4y%5E2%29 Group like terms


3x%283x-2y%29-2y%283x-2y%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of -2y out of the second group


%283x-2y%29%283x-2y%29 Since we have a common term of 3x-2y, we can combine like terms

So 9x%5E2-6xy-6xy%2B4y%5E2 factors to %283x-2y%29%283x-2y%29


So this also means that 9x%5E2-12xy%2B4y%5E2 factors to %283x-2y%29%283x-2y%29 (since 9x%5E2-12xy%2B4y%5E2 is equivalent to 9x%5E2-6xy-6xy%2B4y%5E2)


note: %283x-2y%29%283x-2y%29 is equivalent to %283x-2y%29%5E2 since the term 3x-2y occurs twice. So 9x%5E2-12xy%2B4y%5E2 also factors to %283x-2y%29%5E2



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So our expression goes from 3%289x%5E2-12xy%2B4y%5E2%29 and factors further to 3%283x-2y%29%5E2


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Answer:

So 27x%5E2-36xy%2B12y%5E2 factors to 3%283x-2y%29%5E2