Question

Long divide `x^5+7` by `x^3-1` and find quotient and remainder and express it as fraction?

Answer 1

`(x^5+7)/(x^3-1)=color(magenta)(x^2)+color(blue)(x^2+7)/(x^3-1)`

Explanation:

We use long division method
`color(white)(XXXXXXX)x^2color(white)(XXXXXX)`
`" "x^3-1)bar(x^5color(white)(XXXXXX)+7)`
`color(magenta)(x^2)(x^3-1) ->""ul(x^5-x^2) larr" Subtract"`
`" "0 +x^2color(white)(XXXX)+7`

As the degree of `x^2+7` is less than that of `x^3-1`, we cannot divide further and

Hence, quotient is `x^2` and remainder is `x^2+7` and

`(x^5+7)/(x^3-1)=color(magenta)(x^2)+color(blue)(x^2+7)/(x^3-1)`