SOLUTION: Simplify the rational expression (x^3)-(2x^2)+x-2 ----------- ---- (<- symbolizes dividing side) (3x^2)-3x-8 Im terrible at factoring... Can you help?

Algebra ->  Rational-functions -> SOLUTION: Simplify the rational expression (x^3)-(2x^2)+x-2 ----------- ---- (<- symbolizes dividing side) (3x^2)-3x-8 Im terrible at factoring... Can you help?      Log On


   

Question 129673This question is from textbook McDougal Littell Algebra 2
: Simplify the rational expression
(x^3)-(2x^2)+x-2
---------------- (<- symbolizes dividing side)
(3x^2)-3x-8
Im terrible at factoring... Can you help? This question is from textbook McDougal Littell Algebra 2

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

x%5E3-2x%5E2%2Bx-2 Start with the given expression

%28x%5E3-2x%5E2%29%2B%28x-2%29 Group like terms


x%5E2%28x-2%29%2B1%28x-2%29 Factor out the GCF x%5E2 out of the first group. Factor out the GCF 1 out of the second group


%28x%5E2%2B1%29%28x-2%29 Since we have the common term x-2, we can combine like terms

So the numerator x%5E3-2x%5E2%2Bx-2 factors to %28x%5E2%2B1%29%28x-2%29



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Are you sure that the denominator is 3x%5E2-3x-8? This does not factor

However, 3x%5E2-2x-8 does

Now let's factor the denominator 3x%5E2-2x-8



Looking at 3x%5E2-2x-8 we can see that the first term is 3x%5E2 and the last term is -8 where the coefficients are 3 and -8 respectively.

Now multiply the first coefficient 3 and the last coefficient -8 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient -2? Let's list all of the factors of -24:



Factors of -24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24
(1)*(-24)
(2)*(-12)
(3)*(-8)
(4)*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

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


Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From this list we can see that 4 and -6 add up to -2 and multiply to -24


Now looking at the expression 3x%5E2-2x-8, replace -2x with 4x%2B-6x (notice 4x%2B-6x adds up to -2x. So it is equivalent to -2x)

3x%5E2%2Bhighlight%284x%2B-6x%29%2B-8


Now let's factor 3x%5E2%2B4x-6x-8 by grouping:


%283x%5E2%2B4x%29%2B%28-6x-8%29 Group like terms


x%283x%2B4%29-2%283x%2B4%29 Factor out the GCF of x out of the first group. Factor out the GCF of -2 out of the second group


%28x-2%29%283x%2B4%29 Since we have a common term of 3x%2B4, we can combine like terms

So 3x%5E2%2B4x-6x-8 factors to %28x-2%29%283x%2B4%29


So this also means that 3x%5E2-2x-8 factors to %28x-2%29%283x%2B4%29 (since 3x%5E2-2x-8 is equivalent to 3x%5E2%2B4x-6x-8)




So 3x%5E2-2x-8 factors to %28x-2%29%283x%2B4%29






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


So our expression goes from

%28x%5E3-2x%5E2%2Bx-2%29%2F%283x%5E2-2x-8%29

to


%28%28x%5E2%2B1%29%28x-2%29%29%2F%28%28x-2%29%283x%2B4%29%29



%28%28x%5E2%2B1%29cross%28x-2%29%29%2F%28cross%28x-2%29%283x%2B4%29%29 Cancel like terms


%28x%5E2%2B1%29%2F%283x%2B4%29 Simplify




-----------------------------------------

Answer:



So %28x%5E3-2x%5E2%2Bx-2%29%2F%283x%5E2-2x-8%29 simplifies to %28x%5E2%2B1%29%2F%283x%2B4%29


In other words, %28x%5E3-2x%5E2%2Bx-2%29%2F%283x%5E2-2x-8%29=%28x%5E2%2B1%29%2F%283x%2B4%29