Question

How do you divide `5x^2 - 6x^3 + 1 + 7x` by `3x - 4`?

Answer 1

`(-6x^3 + 5x^2 + 7x + 1) -: (3x - 4) = -2x^2 - x +1`

with remainder `5`.

Explanation:

First of all, order the terms by the power of `x`. In your case, this means

`- 6 x^3 + 5x^2 + 7x + 1`

for the first term.

Now let me walk you through the polynomial long division.

You are basically doing the following operations:

• Divide the dividend's term with the highest power by the divisor's term with the highest power. In your case, that's `(- 6 x^3) -: (3x) = -2 x^2`

• Multiply the result, in your case `-2x^2`, with the divisor: `(-2x^2) * (3x - 4) = -6x^3 + 8x^2`

• Subtract the result from the last step from your divident: `(-6x^3 + 5x^2 + 7x + 1) - (- 6 x^3 + 8x^2) = - 3x^2 + 7x + 1`

• Now, you can repeat all those steps with the term `-3x^2 + 7x + 1` as a new divident.... etc.

In total, your division should look like this:

`color(white)(xx) (-6x^3 + 5x^2 + 7x + 1) -: (3x - 4) = -2x^2 - x +1`
`- (- 6 x^3 + 8 x^2)`
`color(white)(xx) color(white)(xxxxxxxxx) /`
`color(white)(xxxxxxx) - 3 x^2 + 7x`
`color(white)(xxxxx) -(- 3 x^2 + 4x)`
`color(white)(xxxxxxx) color(white)(xxxxxxxxx) /`
`color(white)(xxxxxxxxxxxxxx) 3x +1`
`color(white)(xxxxxxxxxxxii) -(3x-4)`
`color(white)(xxxxxxxxxxxxx) color(white)(xxxxxxxxx) /`
`color(white)(xxxxxxxxxxxxxxxxxx) 5`

This means that

`(-6x^3 + 5x^2 + 7x + 1) -: (3x - 4) = -2x^2 - x +1`

with remainder `5`.

Or, if you prefer a different notation,

`(-6x^3 + 5x^2 + 7x + 1) -: (3x - 4) = -2x^2 - x +1 + 5 / (3x - 4)`