Question

How do you use synthetic division to divide `2x^3-3x^2-5x-12div x-3`?

Answer 1

`2x^2+3x+4`

Explanation:

The first thing we must do is find the value that makes `x-3` equal to zero. In this case, `x=3`. This will be our divisor.

Now, we set up the problem:
If we have `color(green)(2)x^3color(red)(-3)x^2color(blue)(-5)xcolor(purple)(-12)`

`color(white)(3.)` `|` `color(green)(2)color(white)(3.)color(red)(-3)color(white)(3.)color(blue)(-5)color(white)(3.)color(purple)(-12)`
3`color(white)(.)` `|`______

Now, we bring down the `color(green)(2)` and multiply it by the `3`, like this:
`color(white)(3.)` `|` `color(green)(2)color(white)(3.)color(red)(-3)color(white)(3.)color(blue)(-5)color(white)(3.)color(purple)(-12)`
3`color(white)(.)` `|`__`6`____
`color(white)(.......)color(green)(2)`

Now we add the `color(red)(-3)` to the `6`, which gives us `3`. I'll show you what I mean.

`color(white)(3.)` `|` `color(green)(2)color(white)(3.)color(red)(-3)color(white)(3.)color(blue)(-5)color(white)(3.)color(purple)(-12)`
3`color(white)(.)` `|`__`6`___`9`_`12`__
`color(white)(.......)color(green)(2)` `color(white)(....)3` `color(white)(.......)4` `color(white)(.......)0`

I just did the whole thing, and I hope you can see what I did. I took the sum of the answer and multiplied it to the divisor, `3`. Then I take the product and place it in the next column. Then, I add the column together, and whatever the answer is, I multiply it to the `3`.

Anyways, we take the leftover number as the bottom, the `color(green)(2)color(white)(.)3 color(white)(.)4 color(white)(.)0` and rewrite them as an equation, like this: `2x^2+3x+4`. We can factor this further if we want, but I'm going to stop there.

Answer 2

`(2x^3-3x^2-5x-12) -:( x-3) = 2x^2+3x+4`

Explanation:

This answer uses polynomial long division - for synthetic division, please see the other answer!

We write the polynomial and its divisor down in long division form and work through the normal steps of long division:

Then we guess our first term in the quotient, which should subtract from the first term in the dividend:

After performing the subraction, we a left with our first remainder. We next drop the remaining terms in the dividend and guess our next term in the quotient:

and repeat for the last:

Which confirms that `x-3` is a factor of our polynomial evidenced by the fact that we get a zero remainder. So we conclude that:

`(2x^3-3x^2-5x-12) -: (x-3) = 2x^2+3x+4`