Question

How do you divide `(-4x^3-15x^2-4x-12)/(x-4)`?

Answer 1

The remainder is `=-524` and the quotient is `=-4x^2-31x-128`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `4` `|` `color(white)(aaaa)` `-4` `color(white)(aaaa)` `-15` `color(white)(aaaaaa)` `-4` `color(white)(aaaaa)` `-12`

`color(white)(aaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaaaa)` `-16` `color(white)(aaaa)` `-124` `color(white)(aaaa)` `-512`

`color(white)(aaaaaaaaa)`##_________

`color(white)(aaaaaa)` `|` `color(white)(aaaa)` `-4` `color(white)(aaaa)` `-31` `color(white)(aaaa)` `-128` `color(white)(aaaa)` `color(red)(-524)`

The remainder is `=-524` and the quotient is `=-4x^2-31x-128`

ALSO,

Apply the remainder theorem

When a polynomial `f(x)` is divided by `(x-c)`, we get

`f(x)=(x-c)q(x)+r`

Let `x=c`

Then,

`f(c)=0+r`

Here,

`f(x)=-4x^3-15x^2-4x-12`

Therefore,

`f(4)=-4*4^3-15*4^2-4*4-12`

`=-256-240-16-12`

`=-524`

The remainder is `=-524`