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?

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First factor the numerator:

Start with the given expression

Group like terms


Factor out the GCF out of the first group. Factor out the GCF out of the second group


Since we have the common term , we can combine like terms

So the numerator factors to



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Are you sure that the denominator is ? This does not factor

However, does

Now let's factor the denominator



Looking at we can see that the first term is and the last term is 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 Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-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 , replace with (notice adds up to . So it is equivalent to )




Now let's factor by grouping:


Group like terms


Factor out the GCF of out of the first group. Factor out the GCF of out of the second group


Since we have a common term of , we can combine like terms

So factors to


So this also means that factors to (since is equivalent to )




So factors to






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So our expression goes from



to






Cancel like terms


Simplify




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Answer:



So simplifies to


In other words,