SOLUTION: Factor the expression 4n^3 + 8n^2 - 5n - 10 Factor k^2 + kf - 2f^2 Factor 6g^2 + 11g - 35 I am not sure how to factor these. I am especially "stuck" on the one that use

Algebra ->  Expressions-with-variables -> SOLUTION: Factor the expression 4n^3 + 8n^2 - 5n - 10 Factor k^2 + kf - 2f^2 Factor 6g^2 + 11g - 35 I am not sure how to factor these. I am especially "stuck" on the one that use      Log On


   

Question 153303: Factor the expression 4n^3 + 8n^2 - 5n - 10
Factor k^2 + kf - 2f^2
Factor 6g^2 + 11g - 35

I am not sure how to factor these. I am especially "stuck" on the one that uses kf.
Can you help? Thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1



4n%5E3%2B8n%5E2-5n-10 Start with the given expression


%284n%5E3%2B8n%5E2%29%2B%28-5n-10%29 Group like terms


4n%5E2%28n%2B2%29-5%28n%2B2%29 Factor out the GCF 4n%5E2 out of the first group. Factor out the GCF -5 out of the second group


%284n%5E2-5%29%28n%2B2%29 Since we have the common term n%2B2, we can combine like terms


So 4n%5E3%2B8n%5E2-5n-10 factors to %284n%5E2-5%29%28n%2B2%29




# 2


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

Now multiply the first coefficient 1 and the last coefficient -2 to get -2. Now what two numbers multiply to -2 and add to the middle coefficient 1? Let's list all of the factors of -2:



Factors of -2:
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 -2
(1)*(-2)
(-1)*(2)

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


Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1

First NumberSecond NumberSum
1-21+(-2)=-1
-12-1+2=1



From this list we can see that -1 and 2 add up to 1 and multiply to -2


Now looking at the expression k%5E2%2Bkf-2f%5E2, replace kf with -kf%2B2kf (notice -kf%2B2kf adds up to kf. So it is equivalent to kf)

k%5E2%2Bhighlight%28-kf%2B2kf%29%2B-2f%5E2


Now let's factor k%5E2-kf%2B2kf-2f%5E2 by grouping:


%28k%5E2-kf%29%2B%282kf-2f%5E2%29 Group like terms


k%28k-f%29%2B2f%28k-f%29 Factor out the GCF of k out of the first group. Factor out the GCF of 2f out of the second group


%28k%2B2f%29%28k-f%29 Since we have a common term of k-f, we can combine like terms

So k%5E2-kf%2B2kf-2f%5E2 factors to %28k%2B2f%29%28k-f%29


So this also means that k%5E2%2Bkf-2f%5E2 factors to %28k%2B2f%29%28k-f%29 (since k%5E2%2Bkf-2f%5E2 is equivalent to k%5E2-kf%2B2kf-2f%5E2)



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Answer:
So k%5E2%2Bkf-2f%5E2 factors to %28k%2B2f%29%28k-f%29




# 3



Looking at the expression 6g%5E2%2B11g-35, we can see that the first coefficient is 6, the second coefficient is 11, and the last term is -35.


Now multiply the first coefficient 6 by the last term -35 to get %286%29%28-35%29=-210.


Now the question is: what two whole numbers multiply to -210 (the previous product) and add to the second coefficient 11?


To find these two numbers, we need to list all of the factors of -210 (the previous product).


Factors of -210:
1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210
-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -210.
1*(-210)
2*(-105)
3*(-70)
5*(-42)
6*(-35)
7*(-30)
10*(-21)
14*(-15)
(-1)*(210)
(-2)*(105)
(-3)*(70)
(-5)*(42)
(-6)*(35)
(-7)*(30)
(-10)*(21)
(-14)*(15)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 11:


First NumberSecond NumberSum
1-2101+(-210)=-209
2-1052+(-105)=-103
3-703+(-70)=-67
5-425+(-42)=-37
6-356+(-35)=-29
7-307+(-30)=-23
10-2110+(-21)=-11
14-1514+(-15)=-1
-1210-1+210=209
-2105-2+105=103
-370-3+70=67
-542-5+42=37
-635-6+35=29
-730-7+30=23
-1021-10+21=11
-1415-14+15=1



From the table, we can see that the two numbers -10 and 21 add to 11 (the middle coefficient).


So the two numbers -10 and 21 both multiply to -210 and add to 11


Now replace the middle term 11g with -10g%2B21g. Remember, -10 and 21 add to 11. So this shows us that -10g%2B21g=11g.


6g%5E2%2Bhighlight%28-10g%2B21g%29-35 Replace the second term 11g with -10g%2B21g.


%286g%5E2-10g%29%2B%2821g-35%29 Group the terms into two pairs.


2g%283g-5%29%2B%2821g-35%29 Factor out the GCF 2g from the first group.


2g%283g-5%29%2B7%283g-5%29 Factor out 7 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%282g%2B7%29%283g-5%29 Combine like terms. Or factor out the common term 3g-5

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Answer:


So 6g%5E2%2B11g-35 factors to %282g%2B7%29%283g-5%29.


Note: you can check the answer by FOILing %282g%2B7%29%283g-5%29 to get 6g%5E2%2B11g-35 or by graphing the original expression and the answer (the two graphs should be identical).