Question

How do we use long division in polynomials, say dividing `x^6+x^4+x^2+1` by `x^2+1`?

Answer 1

Please see below.

Explanation:

We can write `x^6+x^4+x^2+1` as `x^6+0x^5+x^4+0x^3+x^2+0x+1`

`" "x^6+0x^5+x^4+0x^3+x^2+0x+1`
`color(magenta)(x^4)(x^2+1) ->" "ul(x^6+0x^5+x^4) larr" Subtract"`
`color(white)(XXXXXXXXXXXXXXX)0+0x^3+x^2+0x+1`
`color(magenta)(+1)(x^2+1)->" "color(white)(xxxxxxxxxxx)ul(x^2+0x+1 )`
`" "0`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Hence, quotient is `x^4+1` and remainder is `0` and

`(x^6+x^4+x^2+x+1)/(x^2+1) = color(magenta)(x^4+1)`

For a more complicated sum see here.