Question

How do you long divide `3x^4 - x^3 - 3x^2 + 7x - 12 div x^2 - 4`?

Answer 1

This is tricky to lay out by typing it, so I have hand written it and posted a photo below. The approach is exactly the same as 'normal' long division with numbers.

Explanation:

The tricky bit with this one is that there is an `x^2` term and a constant (number) term in the divisor, but no `x` term. This leads to 'gaps' in the long division, as you will see in the image below.

I first knew I would need `3x` 'lots' of the divisor, `x^2-4`, to make up the first term, `3x^4`. That yielded `3x^4 - 12x^2`, but no `x^3` term. Subtracting this from the original expression yielded `9x^2`, then I brought down the untouched `-x^3` term to join it.

It took `-x` times the divisor to match this -x^3 term, and this yielded `-x^3 - 4x`.

I won't walk through the rest of the calculation as slowly as I have so far, but hopefully the image will make it clear.

Apologies for the quality of the image.