`(x+3)/(x^2-2x-8)-(x-5)/(x^2-12x+32)` Perform the indicated operation(s) and simplify

Expert Answers
marizi eNotes educator| Certified Educator

To be able to perform the indicated operation(s) on `(x+3)/(x^2-2x-8)-(x-5)/(x^2-12x+32)` , we have to express them as similar fractions.

Apply factoring on each expression on the denominator side.

Let:

 `x^2-2x-8=(x+2)(x-4)`

and

 `x^2-12x+32=(x-4)(x-8) `

Determine the LCD by getting the product of the distinct factors from denominator side of each term.

Thus, `LCD =(x+2)(x-4)(x-8)`

                  `=(x^2-2x-8)(x-8)`

                  `= x^3-2x^2-8x-8x^2+16x+64 `

                   `=x^3-10x^2+8x+64`

Express each term by the LCD. Multiply top and bottom of each term by the missing factor.

First term:

`(x+3)/(x^2-2x-8) =(x+3)/((x+2)(x-4)) `

               ` =(x+3)/((x+2)(x-4))*(x-8)/(x-8)`

               ` =((x-8)(x+3))/((x+2)(x-4)(x-8)) `   

               ` =(x^2-5x-24)/(x^3-10x^2+8x+64)`

Second term:

`(x-5)/(x^2-12x+32) =(x-5)/((x-4)(x-8))`

                  `=(x-5)/((x-4)(x-8)) *(x+2)/(x+2) `

                  `=((x-5)(x+2))/((x-4)(x-8)(x+2))`

                  ` =(x^2-5x+2x-10)/(x^3-10x^2+8x+64)`

                  `=(x^2-3x-10)/(x^3-10x^2+8x+64)`

Applying the equivalent fraction in terms of LCD, we get:

`(x+3)/(x^2-2x-8)-(x-5)/(x^2-12x+32)`

`=(x^2-5x-24)/(x^3-10x^2+8x+64) -(x^2-3x-10)/(x^3-10x^2+8x+64)`

`=((x^2-5x-24) -(x^2-3x-10))/(x^3-10x^2+8x+64)`

`=(x^2-5x-24 -x^2+3x+10)/(x^3-10x^2+8x+64)`

`=(x^2-x^2-5x+3x-24+10)/(x^3-10x^2+8x+64) `

`=(0-2x-14)/(x^3-10x^2+8x+64)` 

`=(-2x-14)/(x^3-10x^2+8x+64) or -(2x+14)/(x^3-10x^2+8x+64) `

Final answer:

`(x+3)/(x^2-2x-8)-(x-5)/(x^2-12x+32)=-(2x+14)/(x^3-10x^2+8x+64)`