Question

How do you divide #(x^4-5x^3+2x^2-4) / (x^3-4x) # using polynomial long division?

Answer 1

`x-5+(6x^2-20x-4)/(x^3-4x)`

Explanation:

Incorporating place keepers such as `0x` so that things line up properly.

`" "x^4-5x^3+2x^2+0x-4`
`color(magenta)(x)(x^3-4x) ->" "ul( x^4+0x^3-4x^2) larr" Subtract"`
`" "0 -5x^3+6x^2+0x-4`
`color(magenta)(-5)(x^3-4x)->" "color(white)()ul(-5x^3+0x^2+20x ) larr" Subtract"`
`" "color(blue)(0 +6x^2-20x-4) larr" Remainder"`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Highest order of `6x^2-20x-4` is 2 from `6x^2`

Highest order of the divisor `x^3-4x`

As the order of the divisor is now the highest we stop and we have to express the remainder in fraction form with the divisor as the denominator giving:

`(x^4-5x^3+2x^2-4) -: (x^3-4x) = color(magenta)(x-5 +(color(blue)(6x^2-20x-4))/(x^3-4x)`