SOLUTION: factor completely, factor GCF first 8t^4+75t^3+27t^2

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Question 207857: factor completely, factor GCF first
8t^4+75t^3+27t^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


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8t%5E4%2B75t%5E3%2B27t%5E2 Start with the given expression.


t%5E2%288t%5E2%2B75t%2B27%29 Factor out the GCF t%5E2.


Now let's try to factor the inner expression 8t%5E2%2B75t%2B27


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Looking at the expression 8t%5E2%2B75t%2B27, we can see that the first coefficient is 8, the second coefficient is 75, and the last term is 27.


Now multiply the first coefficient 8 by the last term 27 to get %288%29%2827%29=216.


Now the question is: what two whole numbers multiply to 216 (the previous product) and add to the second coefficient 75?


To find these two numbers, we need to list all of the factors of 216 (the previous product).


Factors of 216:
1,2,3,4,6,8,9,12,18,24,27,36,54,72,108,216
-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-27,-36,-54,-72,-108,-216


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


These factors pair up and multiply to 216.
1*216 = 216
2*108 = 216
3*72 = 216
4*54 = 216
6*36 = 216
8*27 = 216
9*24 = 216
12*18 = 216
(-1)*(-216) = 216
(-2)*(-108) = 216
(-3)*(-72) = 216
(-4)*(-54) = 216
(-6)*(-36) = 216
(-8)*(-27) = 216
(-9)*(-24) = 216
(-12)*(-18) = 216

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 75:


First NumberSecond NumberSum
12161+216=217
21082+108=110
3723+72=75
4544+54=58
6366+36=42
8278+27=35
9249+24=33
121812+18=30
-1-216-1+(-216)=-217
-2-108-2+(-108)=-110
-3-72-3+(-72)=-75
-4-54-4+(-54)=-58
-6-36-6+(-36)=-42
-8-27-8+(-27)=-35
-9-24-9+(-24)=-33
-12-18-12+(-18)=-30



From the table, we can see that the two numbers 3 and 72 add to 75 (the middle coefficient).


So the two numbers 3 and 72 both multiply to 216 and add to 75


Now replace the middle term 75t with 3t%2B72t. Remember, 3 and 72 add to 75. So this shows us that 3t%2B72t=75t.


8t%5E2%2Bhighlight%283t%2B72t%29%2B27 Replace the second term 75t with 3t%2B72t.


%288t%5E2%2B3t%29%2B%2872t%2B27%29 Group the terms into two pairs.


t%288t%2B3%29%2B%2872t%2B27%29 Factor out the GCF t from the first group.


t%288t%2B3%29%2B9%288t%2B3%29 Factor out 9 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%2B9%29%288t%2B3%29 Combine like terms. Or factor out the common term 8t%2B3


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So t%5E2%288t%5E2%2B75t%2B27%29 then factors further to t%5E2%28t%2B9%29%288t%2B3%29


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Answer:


So 8t%5E4%2B75t%5E3%2B27t%5E2 completely factors to t%5E2%28t%2B9%29%288t%2B3%29.


In other words, 8t%5E4%2B75t%5E3%2B27t%5E2=t%5E2%28t%2B9%29%288t%2B3%29.


Note: you can check the answer by expanding t%5E2%28t%2B9%29%288t%2B3%29 to get 8t%5E4%2B75t%5E3%2B27t%5E2 or by graphing the original expression and the answer (the two graphs should be identical).

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