Question

How do you divide `(x^3+2x^2-2x+3)div(x^2-1)` using long division?

Answer 1

Quotient:`color(white)("x")x+2color(white)("XX")`Remainder:`color(white)("x")(-x+5)`
(see below for long division method)

Explanation:

Set up

`color(white)("XXXXX")underline(color(white)("x"x"X"+2"XXXXXXXX"))`
`x^2-1color(white)("x")")"color(white)("x")x^3color(white)("x")+2x^2color(white)("x")-2xcolor(white)("x")+3`

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`x^2` goes into `x^3color(white)("XX")x` times

`color(white)("XXXXX")underline(color(white)("x")xcolor(white)("X")color(white)(+2)color(white)("XXXXXXXX"))`
`x^2-1color(white)("x")")"color(white)("x")x^3color(white)("x")+2x^2color(white)("x")-2xcolor(white)("x")+3`

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Multiply divisor (`x^2-1`) by `x` and subtract from dividend

`color(white)("XXXXX")underline(color(white)("x")xcolor(white)("X")color(white)(+2)color(white)("XXXXXXXX"))`
`x^2-1color(white)("x")")"color(white)("x")x^3color(white)("x")+2x^2color(white)("x")-2xcolor(white)("x")+3`
`color(white)("XXXXXX")underline(x^3color(white)("XXXXX")-xcolor(white)("XXXX")`
`color(white)("XXXXXXxXXX")2x^2color(white)("xx")-xcolor(white)("x")+3`

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Repeating the above process
`x^2` goes into `2x^2color(white)("XXX")+2` times
Multiply (`x^2-1`) by `(+2)` and Subtract

`color(white)("XXXXX")underline(color(white)("x")xcolor(white)("X")+2color(white)("XXXXXXXX"))`
`x^2-1color(white)("x")")"color(white)("x")x^3color(white)("x")+2x^2color(white)("x")-2xcolor(white)("x")+3`
`color(white)("XXXXXX")underline(x^3color(white)("XXXXX")-xcolor(white)("XXXX")`
`color(white)("XXXXXXxXXX")2x^2color(white)("xx")-xcolor(white)("x")+3`
`color(white)("XXXXXXxXXX")underline(2x^2color(white)("XXXXX")-2)`
`color(white)("XXXXXXxXXXXXXX")-xcolor(white)("x")+5`