Question

How do you divide `(8x^3+5x^2-12x+10)/(x^2-3)`?

Answer 1

`(8x^3+5x^2-12x+10)/(x^2-3) = 8x+5 + (12x+25)/(x^2-3)`

Explanation:

I like to long divide just the coefficients, not forgetting to include `0`'s for any missing powers of `x`. In our current example that means the missing term in `x` in the divisor...

The process is similar to long division of numbers.

Write the dividend under the bar and the divisor to the left.

Write the first term `color(blue)(8)` of the quotient above the bar, choosing it so that when multiplied by the divisor `1, 0, -3` the product matches the first term of the dividend.

Write the product `8, 0, -24` of `8` and the divisor under the dividend and subtract it to give a remainder `5, 12`.

Bring down the next term from the dividend alongside it, then choose the next term `color(blue)(5)` of the quotient to match the leading term of our running remainder.

Write the product `5, 0, -15` under the running remainder and subtract it to give the final remainder `12, 25`.

There are no more terms to bring down from the dividend and the remainder is now shorter than the divisor, so this is where we stop.

Our quotient is `8, 5`, meaning `8x+5` and our final remainder is `12, 25` meaning `12x+25`

So:

`(8x^3+5x^2-12x+10)/(x^2-3) = 8x+5 + (12x+25)/(x^2-3)`