Question

How do you find the quotient of `(6x^3+16x^2-60x+39)div(2x+10)`?

Answer 1

The quotient is `=(3x^2-7x+5)`

Explanation:

We can perform either a long division or a synthetic division

Let `f(x)=6x^3+16x^2-60x+39`

Let's perform the synthetic division

`color(white)(aaaa)` `-5` `color(white)(aaaaa)` `|` `color(white)(aaaa)` `6` `color(white)(aaaaaa)` `16` `color(white)(aaaaa)` `-60` `color(white)(aaaaa)` `39`
`color(white)(aaaaaaaaaaaa)`_________

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaaa)` `-30` `color(white)(aaaaaa)` `70` `color(white)(aaaaa)` `-50`
`color(white)(aaaaaaaaaaaa)`________

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `6` `color(white)(aaaaaa)` `-14` `color(white)(aaaaaa)` `10` `color(white)(aaaa)` `color(red)(-11)`

`f(x)=(6x^2-14x+10)-11/(x+5)`

or

`f(x)=3x^2-7x+5-11/(2x+10)`

The remainder is `=-11`

Answer 2

`color(blue)(3x^2-7x+5` and a remainder of `color(blue)(-11`

Explanation:

`color(white)(...............................)color(blue)(3x^2-7x+5`
`color(white)(a)2x+10` `|` `color(white)(.)overline(6x^3+16x^2-60x+39`
`color(white)(...................)ul(6x^3+30x^2)`
`color(white)(.........................)-14x^2-60x`
`color(white)(...........................)ul(-14x^2-70x)`
`color(white)(.........................................)10x+39`
`color(white)(..........................................)ul(10x+50)`
`color(white)(..............................................)0-11`