SOLUTION: factor:4k^2-16kv+16v^2

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Question 168972: factor:4k^2-16kv+16v^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

4k%5E2-16kv%2B16v%5E2 Start with the given expression


4%28k%5E2-4kv%2B4v%5E2%29 Factor out the GCF 4


Now let's focus on the inner expression k%5E2-4kv%2B4v%5E2




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Looking at 1k%5E2-4kv%2B4v%5E2 we can see that the first term is 1k%5E2 and the last term is 4v%5E2 where the coefficients are 1 and 4 respectively.

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



Factors of 4:
1,2

-1,-2 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 4
1*4
2*2
(-1)*(-4)
(-2)*(-2)

note: remember two negative numbers multiplied together make a positive 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
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4



From this list we can see that -2 and -2 add up to -4 and multiply to 4


Now looking at the expression 1k%5E2-4kv%2B4v%5E2, replace -4kv with -2kv%2B-2kv (notice -2kv%2B-2kv adds up to -4kv. So it is equivalent to -4kv)

1k%5E2%2Bhighlight%28-2kv%2B-2kv%29%2B4v%5E2


Now let's factor 1k%5E2-2kv-2kv%2B4v%5E2 by grouping:


%281k%5E2-2kv%29%2B%28-2kv%2B4v%5E2%29 Group like terms


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


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

So 1k%5E2-2kv-2kv%2B4v%5E2 factors to %28k-2v%29%28k-2v%29


So this also means that 1k%5E2-4kv%2B4v%5E2 factors to %28k-2v%29%28k-2v%29 (since 1k%5E2-4kv%2B4v%5E2 is equivalent to 1k%5E2-2kv-2kv%2B4v%5E2)


note: %28k-2v%29%28k-2v%29 is equivalent to %28k-2v%29%5E2 since the term k-2v occurs twice. So 1k%5E2-4kv%2B4v%5E2 also factors to %28k-2v%29%5E2



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So our expression goes from 4%28k%5E2-4kv%2B4v%5E2%29 and factors further to 4%28k-2v%29%5E2


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Answer:

So 4k%5E2-16kv%2B16v%5E2 factors to 4%28k-2v%29%5E2