Question

What is the quotient `(3x^4-4x^2+ 8x-1)div (x-2)`?

Answer 1

`3x^3+6x^2+8x+24 + (47)/(x-2)`

Explanation:

We set up the long division of a polynomial by a simple monomial like this:

`(x-2))bar(3x^4-4x^2+8x-1)`

It works just like the long (numerical) division most of us learned back in elementary school, except now we're dividing with variables.

First we check: how many times does our leading `x` term in the divisor, in this case just `x` (coefficient of `1`), go into our leading `x` term in the dividend, `3x^4`? We would have to multiply `x` by `3x^3` to get `3x^4`. Just like in long division with integers, we put this above the bar.

`color(white)(SPACE)3x^3`
`(x-2))bar(3x^4-4x^2+8x-1)`

Now, we multiply `3x^3` by the divisor and subtract that from `3x^4`.

`color(white)(SPACE)3x^3`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^3(x-2)`

`=>`

`color(white)(SPACE)3x^3`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`

We can subtract `3x^4` from `3x^4` to get `0`. We see that we have no `x^3` term in the dividend, so we treat this as though we're adding `6x^3` to `0`.

Now, we check: how many times does our leading coefficient `x` in the divisor go into `6x^3`? We would have to multiply `x` by `6x^2` to get `6x^3`. Similarly to above:

`=>`

`color(white)(SPACE)3x^3+6x^2`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACE)-6x^2(x-2)`

`=>`

`color(white)(SPACE)3x^3+6x^2`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACES)-6x^3+12x^2`

This gets rid of our `x^3` term, but now we have an `x^2` term. We also have an `x^2` term in the dividend, so we can add this to the current remainder.

`=>`

`color(white)(SPACE)3x^3+6x^2`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPA)+12x^2`

`=>`

`color(white)(SPACE)3x^3+6x^2`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`

Next we check: how many times does our leading coefficient `x` in the divisor go into `8x^2`? We would have to multiply `x` by `8x` to get `8x^2`.

`=>`

`color(white)(SPACE)3x^3+6x^2+8x`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`
`color(white)(SPACESP)-8x(x-2)`

`=>`

`color(white)(SPACE)3x^3+6x^2+8x`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`
`color(white)(SPACESP)-8x^2+16x`

`=>`

`color(white)(SPACE)3x^3+6x^2+8x`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`
`color(white)(SPACESPACES)+16x`

Now we add `16x` to the `8x` term in the dividend.

`=>`

`color(white)(SPACE)3x^3+6x^2+8x`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`
`color(white)(SPACESPACESPA)24x`

Next we check: how many times does our leading coefficient `x` in the divisor go into `24x`? We would have to multiply `x` by `24` to get `24x`.

`=>`

`color(white)(SPACE)3x^3+6x^2+8x+24`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`
`color(white)(SPACESPACESPA)24x`
`color(white)(SPACESPACE)-24(x-2)`

`=>`

`color(white)(SPACE)3x^3+6x^2+8x+24`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`
`color(white)(SPACESPACESPA)24x`
`color(white)(SPACESPACE)-24x+48`

`=>`

`color(white)(SPACE)3x^3+6x^2+8x+24`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`
`color(white)(SPACESPACESPA)24x`
`color(white)(SPACESPACESPACE)48`

Now we add `48` to the `-1` in our dividend.

`=>`

`color(white)(SPACE)3x^3+6x^2+8x+24`
`(x-2))bar(3x^4-4x^2+8x-1)`
`color(white)(SPA)-3x^4+6x^3`
`color(white)(SPACESPA)6x^3`
`color(white)(SPACESPAC)8x^2`
`color(white)(SPACESPACESPA)24x`
`color(white)(SPACESPACESPACE)47`

And of course, `x` goes into `47` zero times. This leaves us with a remainder.

The final answer is therefore `3x^3+6x^2+8x+24 + (47)/(x-2)`