SOLUTION: I got myself lost some place and need help getting back on track. I am factoring completly by grouping. Here are three problems: 1. {{{n(x-y)+(n-1)*(y-x)}}} 2. {{{(a^2+1

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I got myself lost some place and need help getting back on track. I am factoring completly by grouping. Here are three problems: 1. {{{n(x-y)+(n-1)*(y-x)}}} 2. {{{(a^2+1      Log On


   

Question 540990: I got myself lost some place and need help getting back on track.
I am factoring completly by grouping. Here are three problems:
1.
n%28x-y%29%2B%28n-1%29%2A%28y-x%29
2.
%28a%5E2%2B1%29%5E2+-+7%28a%5E2%2B1%29%2B10
3.
%28a%5E2%2B2a%29%5E2+-+2%28a%5E2%2B2a%29+-+3
The final answer to number two is %28a%2B2%29%28a-2%29%28a%2B1%29%28a-1%29 I am just having trouble getting thru the steps.
I understand you can substitute a single varable for a common part of the problem.
So in number2,
b=%28x%5E2%2B1%29
would be:
b%5E2-7b%2B10
Would factor to:
%28b-5%29%28b-2%29
Then putting back in (a^2+1)
%28%28%28a%5E2%2B1%29-5%29%28%28a%5E2%2B1%29-2%29%29
Would further factor to:
%28a%5E2-4%29%28a%5E2-1%29
or
%28a-2%29%5E2+%28a-1%29%5E2
That would be:
%28a%2B2%29%28a-2%29%28a%2B1%29%28a-1%29
which is the answer in the back of the book. But how do you do it by factoring by grouping?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you're wondering how to factor b%5E2-7b%2B10 to get %28b-5%29%28b-2%29



Looking at the expression b%5E2-7b%2B10, we can see that the first coefficient is 1, the second coefficient is -7, and the last term is 10.


Now multiply the first coefficient 1 by the last term 10 to get %281%29%2810%29=10.


Now the question is: what two whole numbers multiply to 10 (the previous product) and add to the second coefficient -7?


To find these two numbers, we need to list all of the factors of 10 (the previous product).


Factors of 10:
1,2,5,10
-1,-2,-5,-10


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


These factors pair up and multiply to 10.
1*10 = 10
2*5 = 10
(-1)*(-10) = 10
(-2)*(-5) = 10

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -7:


First NumberSecond NumberSum
1101+10=11
252+5=7
-1-10-1+(-10)=-11
-2-5-2+(-5)=-7



From the table, we can see that the two numbers -2 and -5 add to -7 (the middle coefficient).


So the two numbers -2 and -5 both multiply to 10 and add to -7


Now replace the middle term -7b with -2b-5b. Remember, -2 and -5 add to -7. So this shows us that -2b-5b=-7b.


b%5E2%2Bhighlight%28-2b-5b%29%2B10 Replace the second term -7b with -2b-5b.


%28b%5E2-2b%29%2B%28-5b%2B10%29 Group the terms into two pairs.


b%28b-2%29%2B%28-5b%2B10%29 Factor out the GCF b from the first group.


b%28b-2%29-5%28b-2%29 Factor out 5 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.


%28b-5%29%28b-2%29 Combine like terms. Or factor out the common term b-2


===============================================================


Answer:


So b%5E2-7b%2B10 factors to %28b-5%29%28b-2%29.


In other words, b%5E2-7b%2B10=%28b-5%29%28b-2%29.


Note: you can check the answer by expanding %28b-5%29%28b-2%29 to get b%5E2-7b%2B10 or by graphing the original expression and the answer (the two graphs should be identical).


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

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim