SOLUTION: t^3 + 10t^2 - 24t, completely factor

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Question 670772: t^3 + 10t^2 - 24t, completely factor
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

t%5E3%2B10t%5E2-24t Start with the given expression.


t%28t%5E2%2B10t-24%29 Factor out the GCF t.


Now let's try to factor the inner expression t%5E2%2B10t-24


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Looking at the expression t%5E2%2B10t-24, we can see that the first coefficient is 1, the second coefficient is 10, and the last term is -24.


Now multiply the first coefficient 1 by the last term -24 to get %281%29%28-24%29=-24.


Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 10?


To find these two numbers, we need to list all of the factors of -24 (the previous product).


Factors of -24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


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


These factors pair up and multiply to -24.
1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 10:


First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From the table, we can see that the two numbers -2 and 12 add to 10 (the middle coefficient).


So the two numbers -2 and 12 both multiply to -24 and add to 10


Now replace the middle term 10t with -2t%2B12t. Remember, -2 and 12 add to 10. So this shows us that -2t%2B12t=10t.


t%5E2%2Bhighlight%28-2t%2B12t%29-24 Replace the second term 10t with -2t%2B12t.


%28t%5E2-2t%29%2B%2812t-24%29 Group the terms into two pairs.


t%28t-2%29%2B%2812t-24%29 Factor out the GCF t from the first group.


t%28t-2%29%2B12%28t-2%29 Factor out 12 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%2B12%29%28t-2%29 Combine like terms. Or factor out the common term t-2


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So t%28t%5E2%2B10t-24%29 then factors further to t%28t%2B12%29%28t-2%29


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Answer:


So t%5E3%2B10t%5E2-24t completely factors to t%28t%2B12%29%28t-2%29.


In other words, t%5E3%2B10t%5E2-24t=t%28t%2B12%29%28t-2%29.


Note: you can check the answer by expanding t%28t%2B12%29%28t-2%29 to get t%5E3%2B10t%5E2-24t or by graphing the original expression and the answer (the two graphs should be identical).