SOLUTION: factor this completely 30t^6 + 25t^5

SOLUTION: factor this completely 30t^6 + 25t^5 - 30t^4

Question 282672: factor this completely 30t^6 + 25t^5 - 30t^4
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Start with the given expression

Factor out the GCF

Now let's focus on the inner expression

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Looking at we can see that the first term is and the last term is where the coefficients are 6 and -6 respectively.

Now multiply the first coefficient 6 and the last coefficient -6 to get -36. Now what two numbers multiply to -36 and add to the middle coefficient 5? Let's list all of the factors of -36:

Factors of -36:
1,2,3,4,6,9,12,18

-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -36
(1)*(-36)
(2)*(-18)
(3)*(-12)
(4)*(-9)
(-1)*(36)
(-2)*(18)
(-3)*(12)
(-4)*(9)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5

First Number	Second Number	Sum
1	-36	1+(-36)=-35
2	-18	2+(-18)=-16
3	-12	3+(-12)=-9
4	-9	4+(-9)=-5
-1	36	-1+36=35
-2	18	-2+18=16
-3	12	-3+12=9
-4	9	-4+9=5

From this list we can see that -4 and 9 add up to 5 and multiply to -36

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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So our expression goes from and factors further to

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Answer:

So completely factors to

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