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.
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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)`