SOLUTION: 35y^3-60y^2-20y Factor the trinomial completely

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Question 191479: 35y^3-60y^2-20y
Factor the trinomial completely
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

35y%5E3-60y%5E2-20y Start with the given expression


5y%287y%5E2-12y-4%29 Factor out the GCF 5y


Now let's focus on the inner expression 7y%5E2-12y-4




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Looking at 7y%5E2-12y-4 we can see that the first term is 7y%5E2 and the last term is -4 where the coefficients are 7 and -4 respectively.

Now multiply the first coefficient 7 and the last coefficient -4 to get -28. Now what two numbers multiply to -28 and add to the middle coefficient -12? Let's list all of the factors of -28:



Factors of -28:
1,2,4,7,14,28

-1,-2,-4,-7,-14,-28 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -28
(1)*(-28)
(2)*(-14)
(4)*(-7)
(-1)*(28)
(-2)*(14)
(-4)*(7)

note: remember, the product of a negative and a positive number is a negative 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
1-281+(-28)=-27
2-142+(-14)=-12
4-74+(-7)=-3
-128-1+28=27
-214-2+14=12
-47-4+7=3



From this list we can see that 2 and -14 add up to -12 and multiply to -28


Now looking at the expression 7y%5E2-12y-4, replace -12y with 2y-14y (notice 2y-14y adds up to -12y. So it is equivalent to -12y)

7y%5E2%2Bhighlight%282y-14y%29-4


Now let's factor 7y%5E2%2B2y-14y-4 by grouping:


%287y%5E2%2B2y%29%2B%28-14y-4%29 Group like terms


y%287y%2B2%29-2%287y%2B2%29 Factor out the GCF of y out of the first group. Factor out the GCF of -2 out of the second group


%28y-2%29%287y%2B2%29 Since we have a common term of 7y%2B2, we can combine like terms

So 7y%5E2%2B2y-14y-4 factors to %28y-2%29%287y%2B2%29


So this also means that 7y%5E2-12y-4 factors to %28y-2%29%287y%2B2%29 (since 7y%5E2-12y-4 is equivalent to 7y%5E2%2B2y-14y-4)



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So our expression goes from 5y%287y%5E2-12y-4%29 and factors further to 5y%28y-2%29%287y%2B2%29


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Answer:

So 35y%5E3-60y%5E2-20y completely factors to 5y%28y-2%29%287y%2B2%29