Question

How do you use polynomial long division to divide `(-x^5+7x^3-x)div(x^3-x^2+1)` and write the polynomial in the form `p(x)=d(x)q(x)+r(x)`?

Answer 1

Please see the explanation below

Explanation:

Perform the long division

`color(white)(aaaa)` `-x^5+0x^4+7x^3+0x^2-x` `color(white)(aaaa)` `|` `x^3-x^2+1`

`color(white)(aaaa)` `-x^5+x^4+0x^3-x^2` `color(white)(aaaaaaaaa)` `|` `-x^2-x`

`color(white)(aaaaaa)` `0-x^4+7x^3+x^2-x`

`color(white)(aaaaaaaa)` `-x^4+x^3+0x^2-x`

`color(white)(aaaaaaaaa)` `-0+6x^3+x^2+0x`

Therefore,

`-x^5+0x^4+7x^3+0x^2-x=(x^3-x^2+1)(-x^2-x)+6x^3+x^2`

The remainder is `r(x)=+6x^3+x^2` and the quotient is `q(x)=-x^2-x`