SOLUTION: Hello. I'm needing some help factoring out this trinomial x^3-60xy^2+7x^2y Thanks

Algebra ->  Equations -> SOLUTION: Hello. I'm needing some help factoring out this trinomial x^3-60xy^2+7x^2y Thanks       Log On


   

Question 184675: Hello.
I'm needing some help factoring out this trinomial
x^3-60xy^2+7x^2y
Thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E3-60xy%5E2%2B7x%5E2y Start with the given expression


x%5E3%2B7x%5E2y-60xy%5E2 Rearrange the terms.


x%28x%5E2%2B7xy-60y%5E2%29 Factor out the GCF x


Now let's focus on the inner expression x%5E2%2B7xy-60y%5E2




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



Looking at 1x%5E2%2B7xy-60y%5E2 we can see that the first term is 1x%5E2 and the last term is -60y%5E2 where the coefficients are 1 and -60 respectively.

Now multiply the first coefficient 1 and the last coefficient -60 to get -60. Now what two numbers multiply to -60 and add to the middle coefficient 7? Let's list all of the factors of -60:



Factors of -60:
1,2,3,4,5,6,10,12,15,20,30,60

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -60
(1)*(-60)
(2)*(-30)
(3)*(-20)
(4)*(-15)
(5)*(-12)
(6)*(-10)
(-1)*(60)
(-2)*(30)
(-3)*(20)
(-4)*(15)
(-5)*(12)
(-6)*(10)

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


Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7

First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4



From this list we can see that -5 and 12 add up to 7 and multiply to -60


Now looking at the expression 1x%5E2%2B7xy-60y%5E2, replace 7xy with -5xy%2B12xy (notice -5xy%2B12xy adds up to 7xy. So it is equivalent to 7xy)

1x%5E2%2Bhighlight%28-5xy%2B12xy%29%2B-60y%5E2


Now let's factor 1x%5E2-5xy%2B12xy-60y%5E2 by grouping:


%281x%5E2-5xy%29%2B%2812xy-60y%5E2%29 Group like terms


x%28x-5y%29%2B12y%28x-5y%29 Factor out the GCF of x out of the first group. Factor out the GCF of 12y out of the second group


%28x%2B12y%29%28x-5y%29 Since we have a common term of x-5y, we can combine like terms

So 1x%5E2-5xy%2B12xy-60y%5E2 factors to %28x%2B12y%29%28x-5y%29


So this also means that 1x%5E2%2B7xy-60y%5E2 factors to %28x%2B12y%29%28x-5y%29 (since 1x%5E2%2B7xy-60y%5E2 is equivalent to 1x%5E2-5xy%2B12xy-60y%5E2)



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




So our expression goes from x%28x%5E2%2B7xy-60y%5E2%29 and factors further to x%28x%2B12y%29%28x-5y%29


------------------
Answer:

So x%5E3-60xy%5E2%2B7x%5E2y factors to x%28x%2B12y%29%28x-5y%29