Question

How do you divide `(x^2-5x-5x^3+x^4)div(5+x)` using synthetic division?

Answer 1

The remainder is `=1300` and the quotient is `=x^3-10x^2+51x-260`

Explanation:

Rearrange the polynomials in decreasing powers of `x`

Let's do the long division

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

`color(white)(aaaa)` `x^4+5x^3` `color(white)(aaaaaaaaaaaaa)` `∣` `x^3-10x^2+51x-260`

`color(white)(aaaa)` `0-10x^3+x^2`

`color(white)(aaaaaa)` `-10x^3-50x^2`

`color(white)(aaaaaaaaaa)` `0+51x^2-5x`

`color(white)(aaaaaaaaaaaa)` `+51x^2+255x`

`color(white)(aaaaaaaaaaaaaaaa)` `0-260x`

`color(white)(aaaaaaaaaaaaaaaaaa)` `-260x-1300`

`color(white)(aaaaaaaaaaaaaaaaaaaaaaa)` `0+1300`

You can use the remainder theorem

`f(x)=x^4-5x^3+x^2-5x`

`f(-5)=625+625+25+25=1300`