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Question
Identify rational expression should be subtracted from `(x^2 + 6x + 8)/(x^3 + 8)` to get `3/(x^2 - 2x + 4)`
Solution
Let p(x) = `(x^2 + 6x + 8)/(x^3 + 8)`
p(x) - q(x) = `3/(x^2 - 2x + 4)`
`(x^2 + 6x + 8)/(x^3 + 8) - "q"(x) = 3/(x^2 - 2x + 4)`
q(x) = `(x^2 + 6x + 8)/(x^3 + 8) - 3/(x^2 - 2x + 4)`
q(x) = `(x^2 + 6x + 8)/((x + 2)(x^2 - 2x + 4)) - 3/(x^2 - 2x + 4)`
= `((x + 2)(x + 4))/((x + 2)(x^2 - 2x + 4)) - 3/(x^2 - 2x + 4)`
= `(x + 4)/(x^2 - 2x + 4) - 3/(x^2 - 2x + 4)`
= `(x + 4 - 3)/(x^2 - 2x + 4)`
= x + 1
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