SOLUTION: Factor: tsquared+3t-10

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Question 293740: Factor: tsquared+3t-10
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression t%5E2%2B3t-10, we can see that the first coefficient is 1, the second coefficient is 3, and the last term is -10.


Now multiply the first coefficient 1 by the last term -10 to get %281%29%28-10%29=-10.


Now the question is: what two whole numbers multiply to -10 (the previous product) and add to the second coefficient 3?


To find these two numbers, we need to list all of the factors of -10 (the previous product).


Factors of -10:
1,2,5,10
-1,-2,-5,-10


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


These factors pair up and multiply to -10.
1*(-10) = -10
2*(-5) = -10
(-1)*(10) = -10
(-2)*(5) = -10

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 3:


First NumberSecond NumberSum
1-101+(-10)=-9
2-52+(-5)=-3
-110-1+10=9
-25-2+5=3



From the table, we can see that the two numbers -2 and 5 add to 3 (the middle coefficient).


So the two numbers -2 and 5 both multiply to -10 and add to 3


Now replace the middle term 3t with -2t%2B5t. Remember, -2 and 5 add to 3. So this shows us that -2t%2B5t=3t.


t%5E2%2Bhighlight%28-2t%2B5t%29-10 Replace the second term 3t with -2t%2B5t.


%28t%5E2-2t%29%2B%285t-10%29 Group the terms into two pairs.


t%28t-2%29%2B%285t-10%29 Factor out the GCF t from the first group.


t%28t-2%29%2B5%28t-2%29 Factor out 5 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%2B5%29%28t-2%29 Combine like terms. Or factor out the common term t-2


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Answer:


So t%5E2%2B3t-10 factors to %28t%2B5%29%28t-2%29.


In other words, t%5E2%2B3t-10=%28t%2B5%29%28t-2%29.


Note: you can check the answer by expanding %28t%2B5%29%28t-2%29 to get t%5E2%2B3t-10 or by graphing the original expression and the answer (the two graphs should be identical).