SOLUTION: I need help factoring the trinomial t^2-24t+135 I got the answer (t-27) (t-3) but when you use the FOIL method to check you get t^2-30t+135. I think that all I have to do is switc

Algebra ->  Equations -> SOLUTION: I need help factoring the trinomial t^2-24t+135 I got the answer (t-27) (t-3) but when you use the FOIL method to check you get t^2-30t+135. I think that all I have to do is switc      Log On


   

Question 548530: I need help factoring the trinomial t^2-24t+135
I got the answer (t-27) (t-3) but when you use the FOIL method to check you get t^2-30t+135. I think that all I have to do is switch some signs but I don't know how to do that.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression t%5E2-24t%2B135, we can see that the first coefficient is 1, the second coefficient is -24, and the last term is 135.


Now multiply the first coefficient 1 by the last term 135 to get %281%29%28135%29=135.


Now the question is: what two whole numbers multiply to 135 (the previous product) and add to the second coefficient -24?


To find these two numbers, we need to list all of the factors of 135 (the previous product).


Factors of 135:
1,3,5,9,15,27,45,135
-1,-3,-5,-9,-15,-27,-45,-135


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


These factors pair up and multiply to 135.
1*135 = 135
3*45 = 135
5*27 = 135
9*15 = 135
(-1)*(-135) = 135
(-3)*(-45) = 135
(-5)*(-27) = 135
(-9)*(-15) = 135

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -24:


First NumberSecond NumberSum
11351+135=136
3453+45=48
5275+27=32
9159+15=24
-1-135-1+(-135)=-136
-3-45-3+(-45)=-48
-5-27-5+(-27)=-32
-9-15-9+(-15)=-24



From the table, we can see that the two numbers -9 and -15 add to -24 (the middle coefficient).


So the two numbers -9 and -15 both multiply to 135 and add to -24


Now replace the middle term -24t with -9t-15t. Remember, -9 and -15 add to -24. So this shows us that -9t-15t=-24t.


t%5E2%2Bhighlight%28-9t-15t%29%2B135 Replace the second term -24t with -9t-15t.


%28t%5E2-9t%29%2B%28-15t%2B135%29 Group the terms into two pairs.


t%28t-9%29%2B%28-15t%2B135%29 Factor out the GCF t from the first group.


t%28t-9%29-15%28t-9%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.


%28t-15%29%28t-9%29 Combine like terms. Or factor out the common term t-9


===============================================================


Answer:


So t%5E2-24t%2B135 factors to %28t-15%29%28t-9%29.


In other words, t%5E2-24t%2B135=%28t-15%29%28t-9%29.


Note: you can check the answer by expanding %28t-15%29%28t-9%29 to get t%5E2-24t%2B135 or by graphing the original expression and the answer (the two graphs should be identical).