SOLUTION: 4x²-4xy-120y

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Question 182955: 4x²-4xy-120y
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Are you sure that the expression is not 4x%5E2-4xy-120y%5E2 ???




4x%5E2-4xy-120y%5E2 Start with the given expression


4%28x%5E2-xy-30y%5E2%29 Factor out the GCF 4


Now let's focus on the inner expression x%5E2-xy-30y%5E2




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Looking at x%5E2-xy-30y%5E2 we can see that the first term is 1x%5E2 and the last term is -30y%5E2 where the coefficients are 1 and -30 respectively.

Now multiply the first coefficient 1 and the last coefficient -30 to get -30. Now what two numbers multiply to -30 and add to the middle coefficient -1? Let's list all of the factors of -30:



Factors of -30:
1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -30
(1)*(-30)
(2)*(-15)
(3)*(-10)
(5)*(-6)
(-1)*(30)
(-2)*(15)
(-3)*(10)
(-5)*(6)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1

First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1



From this list we can see that 5 and -6 add up to -1 and multiply to -30


Now looking at the expression x%5E2-xy-30y%5E2, replace -xy with 5xy-6xy (notice 5xy-6xy adds up to -xy. So it is equivalent to -xy)

x%5E2%2Bhighlight%285xy-6xy%29-30y%5E2


Now let's factor x%5E2%2B5xy-6xy-30y%5E2 by grouping:


%28x%5E2%2B5xy%29%2B%28-6xy-30y%5E2%29 Group like terms


x%28x%2B5y%29-6y%28x%2B5y%29 Factor out the GCF of x out of the first group. Factor out the GCF of -6y out of the second group


%28x-6y%29%28x%2B5y%29 Since we have a common term of x%2B5y, we can combine like terms

So x%5E2%2B5xy-6xy-30y%5E2 factors to %28x-6y%29%28x%2B5y%29


So this also means that x%5E2-xy-30y%5E2 factors to %28x-6y%29%28x%2B5y%29 (since x%5E2-xy-30y%5E2 is equivalent to x%5E2%2B5xy-6xy-30y%5E2)



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So our expression goes from 4%28x%5E2-xy-30y%5E2%29 and factors further to 4%28x-6y%29%28x%2B5y%29


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Answer:

So 4x%5E2-4xy-120y%5E2 completely factors to 4%28x-6y%29%28x%2B5y%29