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

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

30t%5E6%2B25t%5E5-30t%5E4 Start with the given expression


5t%5E4%286t%5E2%2B5t-6%29 Factor out the GCF 5t%5E4


Now let's focus on the inner expression 6t%5E2%2B5t-6




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Looking at 6t%5E2%2B5t-6 we can see that the first term is 6t%5E2 and the last term is -6 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 NumberSecond NumberSum
1-361+(-36)=-35
2-182+(-18)=-16
3-123+(-12)=-9
4-94+(-9)=-5
-136-1+36=35
-218-2+18=16
-312-3+12=9
-49-4+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 6t%5E2%2B5t-6, replace 5t with -4t%2B9t (notice -4t%2B9t adds up to 5t. So it is equivalent to 5t)

6t%5E2%2Bhighlight%28-4t%2B9t%29%2B-6


Now let's factor 6t%5E2-4t%2B9t-6 by grouping:


%286t%5E2-4t%29%2B%289t-6%29 Group like terms


2t%283t-2%29%2B3%283t-2%29 Factor out the GCF of 2t out of the first group. Factor out the GCF of 3 out of the second group


%282t%2B3%29%283t-2%29 Since we have a common term of 3t-2, we can combine like terms

So 6t%5E2-4t%2B9t-6 factors to %282t%2B3%29%283t-2%29


So this also means that 6t%5E2%2B5t-6 factors to %282t%2B3%29%283t-2%29 (since 6t%5E2%2B5t-6 is equivalent to 6t%5E2-4t%2B9t-6)



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So our expression goes from 5t%5E4%286t%5E2%2B5t-6%29 and factors further to 5t%5E4%282t%2B3%29%283t-2%29


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Answer:

So 30t%5E6%2B25t%5E5-30t%5E4 completely factors to 5t%5E4%282t%2B3%29%283t-2%29