SOLUTION: 3y^2-9y-30

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Question 168143: 3y^2-9y-30
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 3y%5E2-9y-30, we can see that the first coefficient is 3, the second coefficient is -9, and the last term is -30.


Now multiply the first coefficient 3 by the last term -30 to get %283%29%28-30%29=-90.


Now the question is: what two whole numbers multiply to -90 (the previous product) and add to the second coefficient -9?


To find these two numbers, we need to list all of the factors of -90 (the previous product).


Factors of -90:
1,2,3,5,6,9,10,15,18,30,45,90
-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -90.
1*(-90)
2*(-45)
3*(-30)
5*(-18)
6*(-15)
9*(-10)
(-1)*(90)
(-2)*(45)
(-3)*(30)
(-5)*(18)
(-6)*(15)
(-9)*(10)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -9:


First NumberSecond NumberSum
1-901+(-90)=-89
2-452+(-45)=-43
3-303+(-30)=-27
5-185+(-18)=-13
6-156+(-15)=-9
9-109+(-10)=-1
-190-1+90=89
-245-2+45=43
-330-3+30=27
-518-5+18=13
-615-6+15=9
-910-9+10=1



From the table, we can see that the two numbers 6 and -15 add to -9 (the middle coefficient).


So the two numbers 6 and -15 both multiply to -90 and add to -9


Now replace the middle term -9y with 6y-15y. Remember, 6 and -15 add to -9. So this shows us that 6y-15y=-9y.


3y%5E2%2Bhighlight%286y-15y%29-30 Replace the second term -9y with 6y-15y.


%283y%5E2%2B6y%29%2B%28-15y-30%29 Group the terms into two pairs.


3y%28y%2B2%29%2B%28-15y-30%29 Factor out the GCF 3y from the first group.


3y%28y%2B2%29-15%28y%2B2%29 Factor out 15 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%283y-15%29%28y%2B2%29 Combine like terms. Or factor out the common term y%2B2

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Answer:


So 3y%5E2-9y-30 factors to %283y-15%29%28y%2B2%29.


Note: you can check the answer by FOILing %283y-15%29%28y%2B2%29 to get 3y%5E2-9y-30 or by graphing the original expression and the answer (the two graphs should be identical).