Quotients and remainders
A few months ago, I wrote a post about replacing long division with a coefficient-matching process. That’s brilliant for C2, but what happens if you’re looking at a C4 question that wants a quotient and a remainder?
Well, it gets a bit more complicated, that’s what happens. But it’s not that much harder, and certainly no harder that algebraic division.
The way to approach it is to think about what happens when you divide numbers - for example,
A typical question asks you to find the quotient and remainder when you divide
We can draw some immediate conclusions about the quotient and the remainder: the quotient has to be a quadratic expression, because
- The quotient is as complicated as the lead term of the top divided by the lead term of the bottom 1
- The remainder is one order less complicated that what you’re dividing by
That means we can say:
Let’s multiply up the bottom:
Multiplying out:
Yikes. Luckily, we can rattle through that quickly:
That means the quotient,
I’m not saying “thou shalt do it this way” - after all, there’s more than one way to do it. However, if you’re making mistakes with long division, perhaps you’d like to give it a try. Let me know how you get on!
Footnotes:
1. Or, more technically, it’s the difference between the orders of the polynomials.