SOLUTION: what is 2y to the third minus 7y squared minus 15y factored?

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Question 150701: what is 2y to the third minus 7y squared minus 15y factored?
Found 2 solutions by Earlsdon, jim_thompson5910:
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Factor:
2y%5E3-7y%5E2-15y Notice that every term has a factor of y, so you can factor a y.
y%282y%5E2-7y-15%29 Now look at the trinomial in parentheses.
The first term can be factored as 2y%5E2+=+2y%2Ay and the last term has factors of:
-15 = -1*15 or 1(-15)
-15 = -3*5 or 3(-5)
To get the middle coefficient of -7, do 2*(-5)+1(3) = -10+3 = -7
So you can try:
y%282y%5E2-7y-15%29+=+y%282y%2B3%29%28y-5%29...and this works.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

2y%5E3-7y%5E2-15y Start with the given expression


y%282y%5E2-7y-15%29 Factor out the GCF y


Now let's focus on the inner expression 2y%5E2-7y-15




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Looking at 2y%5E2-7y-15 we can see that the first term is 2y%5E2 and the last term is -15 where the coefficients are 2 and -15 respectively.

Now multiply the first coefficient 2 and the last coefficient -15 to get -30. Now what two numbers multiply to -30 and add to the middle coefficient -7? 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 -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
1-301+(-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 3 and -10 add up to -7 and multiply to -30


Now looking at the expression 2y%5E2-7y-15, replace -7y with 3y%2B-10y (notice 3y%2B-10y adds up to -7y. So it is equivalent to -7y)

2y%5E2%2Bhighlight%283y%2B-10y%29%2B-15


Now let's factor 2y%5E2%2B3y-10y-15 by grouping:


%282y%5E2%2B3y%29%2B%28-10y-15%29 Group like terms


y%282y%2B3%29-5%282y%2B3%29 Factor out the GCF of y out of the first group. Factor out the GCF of -5 out of the second group


%28y-5%29%282y%2B3%29 Since we have a common term of 2y%2B3, we can combine like terms

So 2y%5E2%2B3y-10y-15 factors to %28y-5%29%282y%2B3%29


So this also means that 2y%5E2-7y-15 factors to %28y-5%29%282y%2B3%29 (since 2y%5E2-7y-15 is equivalent to 2y%5E2%2B3y-10y-15)



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So our expression goes from y%282y%5E2-7y-15%29 and factors further to y%28y-5%29%282y%2B3%29


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Answer:

So 2y%5E3-7y%5E2-15y factors to y%28y-5%29%282y%2B3%29