Question

How do you use polynomial long division to divide `(5x^4-3x^3+2x^2-1)div(x^2+4)` and write the polynomial in the form `p(x)=d(x)q(x)+r(x)`?

Answer 1

The answer is `5x^4-3x^3+2x^2-1=(5x^2-3x-18)(x^2+4)+(12x+71)`

Explanation:

Let's perform the long division

`color(white)(aaaa)` `5x^4-3x^3+2x^2` `color(white)(aaaa)` `-1` `color(white)(aaaa)` `|` `x^2+4`

`color(white)(aaaa)5x^4color(white)(aaaaa)+20x^2color(white)(aaaaaaaaaaa)|5x^2-3x-18`

`color(white)(aaaaaa)` `0-3x^3-18x^2`

`color(white)(aaaaaaaa)` `-3x^3+` `color(white)(aaa)` `-12x`

`color(white)(aaaaaaaaaa)` `+0-18x^2` `color(white)(aa)` `+12x-1`

`color(white)(aaaaaaaaaaaaa)` `-18x^2` `color(white)(aaaaaa)` `-72`

`color(white)(aaaaaaaaaaaaaa)` `+0+12x` `color(white)(aaaaaa)` `+71`

Therefore,

`5x^4-3x^3+2x^2-1=(5x^2-3x-18)(x^2+4)+(12x+71)`