Question

Use long division to divide x^3+x^2-2x+14 by x+3

Answer 1

First,
`x^3=x^2\cdot x`, and if we multiply
`x+3` by
`x^2` we get

`x^3+3x^2`

Subtracting this from the numerator gives a remainder of

`-2x^2-2x+14`

Next,
`-2x^2=-2x\cdot x`, and if we multiply
`x+3` by
`-2x` we have

`-2x^2-6x`

and subtracting this from the previous remainder, we end up with a new remainder of

`4x+14`

Next,
`4x=4\cdot x`, and if we multiply
`x+3` by
`4` we get

`4x+12`

Subtracting from the previous remainder, we get a new remainder of

`2`

which contains no more factors of
`x`, so we're done.

So,

`(x^3+x^2-2x+14)/(x+3)=x^2-2x+4+\frac2{x+3}`