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

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

Jump to Answer

Start with the given expression.

Factor out the GCF .

Now let's try to factor the inner expression

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

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

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

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

Factors of :
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 .
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 :

First Number	Second Number	Sum
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
-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 and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out 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.

Combine like terms. Or factor out the common term

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

So then factors further to

===============================================================

Answer:

So completely factors to .

In other words, .

Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

Jump to Top

RELATED QUESTIONS
factor completely 8t^3... (answered by user_dude2008)

factor completely:... (answered by jim_thompson5910)

Factor completely:... (answered by user_dude2008,Edwin McCravy)

Factor the following equation completely. Factor out the GCF first. 4x^2-6x-4 (answered by stanbon)

Factor completely 8t^2 + 18t +... (answered by rapaljer)

FACTOR OUT THE GCF FIRST THAn factor br regrouping... (answered by ankor@dixie-net.com)

Factor our the GCF first, then factor by grouping.... (answered by Alan3354)

How would I factor completely... (answered by KMST)

Factor 200t + 8t^3 - 80t^2.... (answered by scott8148)