Question

How do you long divide `(x^4-2x^2+1) div (x-1)`?

Answer 1

`color(green)(x^3+x^2-x-1)`
(see below for long division method)

Explanation:

Write as a long division being sure to include even zero coefficient powers of `x`
`color(white)("XXXxX")underline(color(white)("XXXXXXXXXXXXXXXXXXXX"))`
`x-1 color(white)("X")")"color(white)("X")x^4color(white)("X")+0x^3color(white)("X")-2x^2color(white)("X")+0xcolor(white)("X")+1`

Divide the leading term of the `color(red)("divisor")` into the leading term of the `color(blue)("dividend")` and write `color(green)("quotient")` above the line.
`color(white)("XXXxX")underline(color(white)("X")color(green)(x^3)color(white)("XXXXXXXXXXXXXXXXX"))`
`color(red)(x)-1 color(white)("X")")"color(white)("X")color(blue)(x^4)color(white)("X")+0x^3color(white)("X")-2x^2color(white)("X")+0xcolor(white)("X")+1`

Multiply the `color(green)("quotient term")` just determined and the `color(red)("entire divisor")`; then subtract the `color(brown)("product")` from the `color(blue)("dividend")`
`color(white)("XXXxX")underline(color(white)("X")color(green)(x^3)color(white)("XXXXXXXXXXXXXXXXX"))`
`color(red)(x-1) color(white)("X")")"color(white)("X")color(blue)(x^4color(white)("X")+0x^3)color(white)("X")-2x^2color(white)("X")+0xcolor(white)("X")+1`
`color(white)("XXXXXX")underline(color(brown)(x^4color(white)("X")-x^3))`
`color(white)("XXXXXXXXXX")x^3color(white)("X")-2x^2`

Repeat this process until all terms of the dividend have been processed.
`color(white)("XXXxX")underline(color(white)("X")x^3color(white)("X")+x^2color(white)("XX")-xcolor(white)("XX")-1color(white)("XXXXX"))`
`x-1 color(white)("X")")"color(white)("X")x^4color(white)("X")+0x^3color(white)("X")-2x^2color(white)("X")+0xcolor(white)("X")+1`
`color(white)("XXXXXX")underline(x^4color(white)("X")-x^3)`
`color(white)("XXXXXXXXXX")x^3color(white)("X")-2x^2`
`color(white)("XXXXXXXXXX")underline(x^3color(white)("Xx")-x^2)`
`color(white)("XXXXXXXXXXXxXX")-x^2color(white)("X")+0x`
`color(white)("XXXXXXXXXXXXXx")underline(-x^2color(white)("X")+1x)`
`color(white)("XXXXXXXXXXXXXXxXXXX")-xcolor(white)("X")+1`
`color(white)("XXXXXXXXXXXXXXxXXXX")underline(-xcolor(white)("X")+1`
`color(white)("XXXXXXXXXXXXXXXXXXXxXXXX")0`