How do you divide `(18x^4+9x^3+3x^2)/(3x^2+1)` using synthetic division?

Synthetic division is more commonly used with linear divisors and especially with monic linear divisors. It is possible to use a form of it with divisors of degree `> 1` as in this example, but it may be easier to use in the form of long division of coefficients...

`color(white)(30000010")")underline(color(white)(001)6color(white)(000)3color(white)(00)-1color(white)(00-000-1)`
`3color(white)(00)0color(white)(00)1color(white)(0)")"color(white)(00)18color(white)(000)9color(white)(00-)3color(white)(00-)0color(white)(00-)0`
`color(white)(30000010")"00)underline(18color(white)(000)0color(white)(00-)6)`
`color(white)(30000010")"0000000)9color(white)(00)-3color(white)(00-)0`
`color(white)(30000010")"0000000)underline(9color(white)(00-)0color(white)(00-)3)`
`color(white)(30000010")"0000000900)-3color(white)(00)-3color(white)(00-)0`
`color(white)(30000010")"000000090)underline(color(white)(0)-3color(white)(00-)0color(white)(00)-1)`
`color(white)(30000010")"0000000900-300)-3color(white)(00-)1`

The first important thing to note is the inclusion of `0`'s for missing powers of `x` in both the dividend `18x^4+9x^3+3x^2` represented as `18color(white)(0)9color(white)(0)3color(white)(0)0color(white)(0)0` and the divisor `3x^2+1` represented as `3color(white)(0)0color(white)(0)1`.

The actual process of long division is similar to long division of numbers.

In our example, we arrive at a quotient `6color(white)(0)3color(white)(0)-1` representing `6x^2+3x-1` and remainder `-3color(white)(0)1` representing `-3x+1`.