SOLUTION: Factor 2t^5-14t^4+24t^3

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor 2t^5-14t^4+24t^3      Log On


   

Question 190288: Factor
2t^5-14t^4+24t^3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

2t%5E5-14t%5E4%2B24t%5E3 Start with the given expression


2t%5E3%28t%5E2-7t%2B12%29 Factor out the GCF 2t%5E3


Now let's focus on the inner expression t%5E2-7t%2B12




------------------------------------------------------------




Looking at the expression t%5E2-7t%2B12, we can see that the first coefficient is 1, the second coefficient is -7, and the last term is 12.


Now multiply the first coefficient 1 by the last term 12 to get %281%29%2812%29=12.


Now the question is: what two whole numbers multiply to 12 (the previous product) and add to the second coefficient -7?


To find these two numbers, we need to list all of the factors of 12 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -7:


First NumberSecond NumberSum
1121+12=13
262+6=8
343+4=7
-1-12-1+(-12)=-13
-2-6-2+(-6)=-8
-3-4-3+(-4)=-7



From the table, we can see that the two numbers -3 and -4 add to -7 (the middle coefficient).


So the two numbers -3 and -4 both multiply to 12 and add to -7


Now replace the middle term -7t with -3t-4t. Remember, -3 and -4 add to -7. So this shows us that -3t-4t=-7t.


t%5E2%2Bhighlight%28-3t-4t%29%2B12 Replace the second term -7t with -3t-4t.


%28t%5E2-3t%29%2B%28-4t%2B12%29 Group the terms into two pairs.


t%28t-3%29%2B%28-4t%2B12%29 Factor out the GCF t from the first group.


t%28t-3%29-4%28t-3%29 Factor out 4 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-4%29%28t-3%29 Combine like terms. Or factor out the common term t-3


So t%5E2-7t%2B12 factors to %28t-4%29%28t-3%29.




------------------------------------------------------------




So our expression goes from 2t%5E3%28t%5E2-7t%2B12%29 and factors further to 2t%5E3%28t-4%29%28t-3%29


------------------
Answer:

So 2t%5E5-14t%5E4%2B24t%5E3 completely factors to 2t%5E3%28t-4%29%28t-3%29