SOLUTION: How do you factor 3y^2-12y-135

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Question 439604: How do you factor 3y^2-12y-135
Answer by rwm(914) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


3%2Ay%5E2-12%2Ay-135 Start with the given expression.



3%28y%5E2-4y-45%29 Factor out the GCF 3.



Now let's try to factor the inner expression y%5E2-4y-45



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Looking at the expression y%5E2-4y-45, we can see that the first coefficient is 1, the second coefficient is -4, and the last term is -45.



Now multiply the first coefficient 1 by the last term -45 to get %281%29%28-45%29=-45.



Now the question is: what two whole numbers multiply to -45 (the previous product) and add to the second coefficient -4?



To find these two numbers, we need to list all of the factors of -45 (the previous product).



Factors of -45:

1,3,5,9,15,45

-1,-3,-5,-9,-15,-45



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



These factors pair up and multiply to -45.

1*(-45) = -45
3*(-15) = -45
5*(-9) = -45
(-1)*(45) = -45
(-3)*(15) = -45
(-5)*(9) = -45


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -4:



First NumberSecond NumberSum
1-451+(-45)=-44
3-153+(-15)=-12
5-95+(-9)=-4
-145-1+45=44
-315-3+15=12
-59-5+9=4




From the table, we can see that the two numbers 5 and -9 add to -4 (the middle coefficient).



So the two numbers 5 and -9 both multiply to -45 and add to -4



Now replace the middle term -4y with 5y-9y. Remember, 5 and -9 add to -4. So this shows us that 5y-9y=-4y.



y%5E2%2Bhighlight%285y-9y%29-45 Replace the second term -4y with 5y-9y.



%28y%5E2%2B5y%29%2B%28-9y-45%29 Group the terms into two pairs.



y%28y%2B5%29%2B%28-9y-45%29 Factor out the GCF y from the first group.



y%28y%2B5%29-9%28y%2B5%29 Factor out 9 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.



%28y-9%29%28y%2B5%29 Combine like terms. Or factor out the common term y%2B5



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So 3%28y%5E2-4y-45%29 then factors further to 3%28y-9%29%28y%2B5%29



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Answer:



So 3%2Ay%5E2-12%2Ay-135 completely factors to 3%28y-9%29%28y%2B5%29.



In other words, 3%2Ay%5E2-12%2Ay-135=3%28y-9%29%28y%2B5%29.



Note: you can check the answer by expanding 3%28y-9%29%28y%2B5%29 to get 3%2Ay%5E2-12%2Ay-135 or by graphing the original expression and the answer (the two graphs should be identical).