Question

How do you long divide `( x^5 - 1) div (x - 1)`?

Answer 1

`(x^5-1) div (x-1) = color(blue)(x^4+x^3+x^2+x+1)`
See below for long division method.

Explanation:

Divide the leading term of the divisor into the leading term of the dividend
`color(white)("XXXX")underline(color(white)("XX")color(red)(x^4)color(white)(+x^3+x^2+x^1+1))`
`color(red)(x)-1" ) "x^5color(white)("XXXXXXXxXX") -1`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Multiply the quotient term just calculated by the entire divisor
then subtract from the dividend.
`color(white)("XXXX")underline(color(white)("XX")color(blue)(x^4)color(white)(+x^3+x^2+x^1+1))`
`color(blue)(x-1)" ) "x^5color(white)("XXXXXXXxXX") -1`
`color(white)("XXXXXx")underline(color(blue)(x^5-x^4))`
`color(white)("XXXXXXXX")color(green)(x^4+0x^3)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Repeat this process until the entire dividend has been evaluated.
`color(white)("XXXX")underline(color(white)("XX")x^4+x^3+x^2+x^1+1)`
`x-1" ) "x^5color(white)("XXXXXXXxXX") -1`
`color(white)("XXXXXx")underline(x^5-x^4)`
`color(white)("XXXXXXXX")x^4+0x^3`
`color(white)("XXXXXXXX")underline(x^4-x^3)`
`color(white)("XXXXXXXXXXX")x^3+0x^2`
`color(white)("XXXXXXXXXXX")underline(x^3-x^2)`
`color(white)("XXXXXXXXXXXXXX")x^2-1`
`color(white)("XXXXXXXXXXXXXX")underline(x^2-1)`
`color(white)("XXXXXXXXXXXXXXXXX")0`

`color(white)("XXXX")underline(color(white)("XX")x^4+x^3+x^2+x^1+1)`
`x-1" ) "x^5color(white)("XXXXXXXxXX") -1`
`color(white)("XXXXXx")underline(x^5-x^4)`
`color(white)("XXXXXXXX")x^4+0x^3`
`color(white)("XXXXXXXX")underline(x^4-x^3)`
`color(white)("XXXXXXXXXXX")x^3+0x^2`
`color(white)("XXXXXXXXXXX")underline(x^3-x^2)`
`color(white)("XXXXXXXXXXXXXX")x^2-1`
`color(white)("XXXXXXXXXXXXXX")underline(x^2-1)`
`color(white)("XXXXXXXXXXXXXXXXX")0`