Question

How do you use synthetic division to divide `(12x^4 + 20x^3 - 24x^2 + 20x + 35)` by `3x+5`?

Answer 1

The remainder is `-65` and the quotient is `=4x^3-8x+20`

Explanation:

Let's perform the synthetic division

Divide by `3`

`color(white)(aaaa)` `-5/3` `|` `color(white)(aaaa)` `4` `color(white)(aaaa)` `20/3` `color(white)(aaaaaa)` `-8` `color(white)(aaaaaa)` `20/3` `color(white)(aaaa)` `35/3`

`color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaa)` `-20/3` `color(white)(aaaaaa)` `0` `color(white)(aaaaaaa)` `40/3` `color(white)(aaa)` `-100/3`

`color(white)(aaaaaaaaa)` _________________________________________________________##

`color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `4` `color(white)(aaaaa)` `0` `color(white)(aaaaaaa)` `-8` `color(white)(aaaaaaa)` `20` `color(white)(aaaa)` `color(red)(-65/3)`

The remainder is `-65` and the quotient is `=4x^3+0x^2-8x+20`

`(12x^4+20x^3-24x^2+20x+35)/(3x+5)=4x^3-8x+20-(65/3)/(x+5/3)`

`=4x^3-8x+20-(65)/(3x+5)`