SOLUTION: Factor completely, if prime, state this 5tˆ3 – 21tˆ2 + 18t

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Question 550525: Factor completely, if prime, state this
5tˆ3 – 21tˆ2 + 18t
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

5t%5E3-21t%5E2%2B18t Start with the given expression.


t%285t%5E2-21t%2B18%29 Factor out the GCF t.


Now let's try to factor the inner expression 5t%5E2-21t%2B18


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Looking at the expression 5t%5E2-21t%2B18, we can see that the first coefficient is 5, the second coefficient is -21, and the last term is 18.


Now multiply the first coefficient 5 by the last term 18 to get %285%29%2818%29=90.


Now the question is: what two whole numbers multiply to 90 (the previous product) and add to the second coefficient -21?


To find these two numbers, we need to list all of the factors of 90 (the previous product).


Factors of 90:
1,2,3,5,6,9,10,15,18,30,45,90
-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90


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


These factors pair up and multiply to 90.
1*90 = 90
2*45 = 90
3*30 = 90
5*18 = 90
6*15 = 90
9*10 = 90
(-1)*(-90) = 90
(-2)*(-45) = 90
(-3)*(-30) = 90
(-5)*(-18) = 90
(-6)*(-15) = 90
(-9)*(-10) = 90

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -21:


First NumberSecond NumberSum
1901+90=91
2452+45=47
3303+30=33
5185+18=23
6156+15=21
9109+10=19
-1-90-1+(-90)=-91
-2-45-2+(-45)=-47
-3-30-3+(-30)=-33
-5-18-5+(-18)=-23
-6-15-6+(-15)=-21
-9-10-9+(-10)=-19



From the table, we can see that the two numbers -6 and -15 add to -21 (the middle coefficient).


So the two numbers -6 and -15 both multiply to 90 and add to -21


Now replace the middle term -21t with -6t-15t. Remember, -6 and -15 add to -21. So this shows us that -6t-15t=-21t.


5t%5E2%2Bhighlight%28-6t-15t%29%2B18 Replace the second term -21t with -6t-15t.


%285t%5E2-6t%29%2B%28-15t%2B18%29 Group the terms into two pairs.


t%285t-6%29%2B%28-15t%2B18%29 Factor out the GCF t from the first group.


t%285t-6%29-3%285t-6%29 Factor out 3 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-3%29%285t-6%29 Combine like terms. Or factor out the common term 5t-6


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So t%285t%5E2-21t%2B18%29 then factors further to t%28t-3%29%285t-6%29


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Answer:


So 5t%5E3-21t%5E2%2B18t completely factors to t%28t-3%29%285t-6%29.


In other words, 5t%5E3-21t%5E2%2B18t=t%28t-3%29%285t-6%29.


Note: you can check the answer by expanding t%28t-3%29%285t-6%29 to get 5t%5E3-21t%5E2%2B18t or by graphing the original expression and the answer (the two graphs should be identical).