Question

How do you divide `(x^4+x^2+2) / (x-2)`?

Answer 1

`x^3+2x^2+5x+10+\frac{22}{x-2}`

Explanation:

We have to find what multiplied by `x-2` cancels `x^4`. In this case it is `x^3`. Multiplying by `x^3` we get `x^4-2x^3`. We can then subtract this from the numerator to get
`x^4+x^2+2-(x^4-2x^3)=2x^3+x^2+2`

Now we need to find a value that will cancel `2x^3`. This value is `2x^2`. Multiplying by this we get `2x^3-4x^2`. Subtracting we get
`2x^3+x^2+2-(2x^3-4x^2)=5x^2+2`

The value that will cancel out `5x^2` is `5x`. Subtracting we get
`5x^2+2-(5x^2-10x)=10x+2`

To cancel out `10x` we can us `10`.
`10x+2-(10x-20)=22`

Since we can't divide anymore, 22 is the remainder. We add the remainder divided by the denominator to the rest of the terms we got to get
`x^3+2x^2+5x+10+\frac{22}{x-2}`