Question

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

Answer 1

(see below for steps of long division)
`(a^3-2a^2+a)div(a-1)=color(green)(a^2-a`

Explanation:

Here it is; one step at a time:

Write in long division form:
`color(white)("XXXX")`
`a-1 )bar(a^3-2a^2+a)`

Leading term of divisor into leading term of dividend
`color(white)("XXXX")color(red)(a^2)`
`color(red)(a)-1 )bar(color(red)(a^3)-2a^2+a)`

Leading term of quotient times complete divisor
`color(white)("XXXX")color(red)(a^2)`
`color(red)(a-1) )bar(a^3-2a^2+a)`
`color(white)("XXXX")underline(color(red)(a^3-a^2)`

Subtract to get progressive dividend/remainder
`color(white)("XXXX")a^2`
`a-1) bar(a^3-2a^2+a)`
`color(white)("XXXX")underline(a^3-a^2)`
`color(white)("XXXXXX")color(red)(-a^2+a)`

Leading term of divisor into leading term of progressive dividend/remainder
`color(white)("XXXX")a^2color(red)(-a)`
`color(red)(a)-1) bar(a^3-2a^2+a)`
`color(white)("XXXX")underline(a^3-a^2)`
`color(white)("XXXXXX")color(red)(-a^2)+a`

Next term of quotient times complete divisor
`color(white)("XXXX")a^2color(red)(-a)`
`color(red)(a-1)) bar(a^3-2a^2+a)`
`color(white)("XXXX")underline(a^3-a^2)`
`color(white)("XXXXXX")-a^2+a`
`color(white)("XXXXXX")underline(color(red)(-a^2+a))`

Subtract to get final remainder
`color(white)("XXXX")a^2-a`
`a-1) bar(a^3-2a^2+a)`
`color(white)("XXXX")underline(a^3-a^2)`
`color(white)("XXXXXX")-a^2+a`
`color(white)("XXXXXX")underline(-a^2+a)`
`color(white)("XXXXXXXXXX")color(red)(0)`