SOLUTION: Ok, I'm working on factoring completely. Here is my equation to factor: -4n^4 + 40n^3 - 100n^2 that equals = -4n^2 (n^2 - 10n + 25) and I'm supposed to factor it d

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Ok, I'm working on factoring completely. Here is my equation to factor: -4n^4 + 40n^3 - 100n^2 that equals = -4n^2 (n^2 - 10n + 25) and I'm supposed to factor it d      Log On


   

Question 405542: Ok, I'm working on factoring completely.
Here is my equation to factor:
-4n^4 + 40n^3 - 100n^2
that equals =
-4n^2 (n^2 - 10n + 25)
and I'm supposed to factor it down more so that the answer is
-4n^2(n - 5)^2
I get it all until the last step. How and Why am I supposed to do that?
Mucho thanks for any help!
Found 2 solutions by ewatrrr, jim_thompson5910:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi



-4n^4 + 40n^3 - 100n^2 

-4n^2 (n^2 - 10n + 25)   |good work with this

-4n^2 (n-5)(n-5)    |to complete, the quadratic expression can be factored     

Note:SUM of the inner product(-5n) and the outer product(-5n) = -10n

Finaly, can be written

-4n^2 (n-5)^2

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

-4n%5E4%2B40n%5E3-100n%5E2 Start with the given expression


-4n%5E2%28n%5E2-10n%2B25%29 Factor out the GCF -4n%5E2


Now let's focus on the inner expression n%5E2-10n%2B25




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Looking at n%5E2-10n%2B25 we can see that the first term is n%5E2 and the last term is 25 where the coefficients are 1 and 25 respectively.

Now multiply the first coefficient 1 and the last coefficient 25 to get 25. Now what two numbers multiply to 25 and add to the middle coefficient -10? Let's list all of the factors of 25:



Factors of 25:
1,5

-1,-5 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 25
1*25
5*5
(-1)*(-25)
(-5)*(-5)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -10? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -10

First NumberSecond NumberSum
1251+25=26
555+5=10
-1-25-1+(-25)=-26
-5-5-5+(-5)=-10



From this list we can see that -5 and -5 add up to -10 and multiply to 25


Now looking at the expression n%5E2-10n%2B25, replace -10n with -5n-5n (notice -5n-5n combines back to -10n. So it is equivalent to -10n)

n%5E2%2Bhighlight%28-5n-5n%29%2B25


Now let's factor n%5E2-5n-5n%2B25 by grouping:


%28n%5E2-5n%29%2B%28-5n%2B25%29 Group like terms


n%28n-5%29-5%28n-5%29 Factor out the GCF of n out of the first group. Factor out the GCF of -5 out of the second group


%28n-5%29%28n-5%29 Since we have a common term of n-5, we can combine like terms

So n%5E2-5n-5n%2B25 factors to %28n-5%29%28n-5%29


So this also means that n%5E2-10n%2B25 factors to %28n-5%29%28n-5%29 (since n%5E2-10n%2B25 is equivalent to n%5E2-5n-5n%2B25)


note: %28n-5%29%28n-5%29 is equivalent to %28n-5%29%5E2 since the term n-5 occurs twice. So n%5E2-10n%2B25 also factors to %28n-5%29%5E2



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So our expression goes from -4n%5E2%28n%5E2-10n%2B25%29 and factors further to -4n%5E2%28n-5%29%5E2


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Answer:

So -4n%5E4%2B40n%5E3-100n%5E2 factors to -4n%5E2%28n-5%29%5E2

If you need more help, email me at jim_thompson5910@hotmail.com

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Jim