SOLUTION: Factor the trinomial. v^3 - 4v^2 - 21v

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Question 133006: Factor the trinomial.
v^3 - 4v^2 - 21v

Found 2 solutions by jim_thompson5910, solver91311:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

v%5E3-4v%5E2-21v Start with the given expression


v%28v%5E2-4v-21%29 Factor out the GCF v


Now let's focus on the inner expression v%5E2-4v-21




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Looking at 1v%5E2-4v-21 we can see that the first term is 1v%5E2 and the last term is -21 where the coefficients are 1 and -21 respectively.

Now multiply the first coefficient 1 and the last coefficient -21 to get -21. Now what two numbers multiply to -21 and add to the middle coefficient -4? Let's list all of the factors of -21:



Factors of -21:
1,3,7,21

-1,-3,-7,-21 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -21
(1)*(-21)
(3)*(-7)
(-1)*(21)
(-3)*(7)

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


Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4

First NumberSecond NumberSum
1-211+(-21)=-20
3-73+(-7)=-4
-121-1+21=20
-37-3+7=4



From this list we can see that 3 and -7 add up to -4 and multiply to -21


Now looking at the expression 1v%5E2-4v-21, replace -4v with 3v%2B-7v (notice 3v%2B-7v adds up to -4v. So it is equivalent to -4v)

1v%5E2%2Bhighlight%283v%2B-7v%29%2B-21


Now let's factor 1v%5E2%2B3v-7v-21 by grouping:


%281v%5E2%2B3v%29%2B%28-7v-21%29 Group like terms


v%28v%2B3%29-7%28v%2B3%29 Factor out the GCF of v out of the first group. Factor out the GCF of -7 out of the second group


%28v-7%29%28v%2B3%29 Since we have a common term of v%2B3, we can combine like terms

So 1v%5E2%2B3v-7v-21 factors to %28v-7%29%28v%2B3%29


So this also means that 1v%5E2-4v-21 factors to %28v-7%29%28v%2B3%29 (since 1v%5E2-4v-21 is equivalent to 1v%5E2%2B3v-7v-21)



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So our expression goes from v%28v%5E2-4v-21%29 and factors further to v%28v-7%29%28v%2B3%29


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Answer:

So v%5E3-4v%5E2-21v factors to v%28v-7%29%28v%2B3%29

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
v%5E3+-+4v%5E2+-+21v

To begin with, there is at least one 'v' in every term, so factor that out:
v%28v%5E2-4v-21%29

Now we know that 3%2A-7=-21 and 3-7=-4, so:
v%5E2-4v-21=%28v%2B3%29%28v-7%29

Putting it all together:
v%5E3+-+4v%5E2+-+21v=v%28v%2B3%29%28v-7%29