Solve with long division

Polynomial long division is no different than regular long division. We begin by rewriting the original question into a more traditional division format:

(3x^4 – 6x^3 – 5x + 10) / (x – 2)

___________________

x – 2 | 3x^4 – 6x^3 – 5x + 10

Now that we have re-written the problem we can solve the problem by identifying what divisor (x – 2) can be multiplied by to solve:

_3x^3______________

x – 2 | 3x^4 – 6x^3 – 5x + 10

The first step is to find what the divisor can be multiplied by to equal the first part of the dividend (3x^4 – 6x^3). By multiplying x – 2 * 3x^3 we get 3x^4 – 6x^3, which we will subtract from the dividend leaving no remainder:

_3x^3______________

x – 2 | 3x^4 – 6x^3 – 5x + 10

3x^4 – 6x^3

0

We now bring down the remaining portion of the dividend (-5x + 10) and then multiply x – 2 by -5:

_3x^3______- 5________

x – 2 | 3x^4 – 6x^3 – 5x + 10

3x^4 – 6x^3 ↓

0 – 5x + 10

– 5x + 10

0

Once again there is no remainder, therefore the answer is 3x^3 – 5.

For the sake of argument, let us say there was a remainder. The solution would then be 3x^3 – 5 + the remained over the original divisor.