The Euclidian division algorithm

We’re now going to learn how to divide one polynomial by another. This is definitely more fun with algebra. Polynomial division, well, there’s a classical algorithm called Euclidean division that allows you to divide a polynomial P of x by another polynomial D of x, provided that the degree of D is no greater than that of P. The algorithm will yield an answer of the form P over D. It is another polynomial Q plus another one R over D. Q will be called the quotient, and the point of the remainder R is that it’s of degree strictly less than the degree of D. If you multiply across by D, you can write the answer in the form P equals DQ plus R.

To illustrate the procedure then, we’re going to apply this long division, as it’s called, to the case of a polynomial of degree 5, which we’re dividing by a quadratic polynomial. And here’s the setup. You give yourself a scheme like this, where you’re going to put the denominator, the divisor, on the right and the dividend on the left. Now, in some countries, like where I grew up, you put the divisor on the left, instead of on the right. This makes, really, very little difference to the procedure. One can get used to either choice. So the divisor you put here. That’s the denominator, D. The dividend, as it’s called, is the numerator. You put it on the left.

Now, do you see that, in writing the numerator, I have included 0 times x to the fourth? That may seem a little odd. It doesn’t change anything, but the presence of that term will help us put the powers of x in the right place during the algorithm, as we’ll see. We’re now ready to begin the first iteration. We take the highest order term of the divisor, that’s x squared, and we divide it into the higher order term of the dividend, that’s x to the fifth. The result is x cubed. We carefully record the x cubed, and we multiply the x cubed, now, by every term in the divisor, the three terms. That will give us three terms on the left.

We draw a line, we do a subtraction, and we bring down, from the dividend, the next term. That was one iteration, and now we’re ready to start again. How do we start? With the same x squared, but now divided into the minus x to the fourth. That gives us minus x squared, which we record. We multiply that by all the terms in the divisor. We get the three terms on the left. We draw a line. We subtract. We bring down the next term from the dividend. That’s another iteration. We continue. The x squared does divide into the minus 2 x cubed, and it gives us minus 2x. We multiply that by all the terms in the divisor. We get this.

We subtract, and we finally get that, and we bring down the plus 3. And then similarly, the next iteration gives us a plus 1, and now after we’ve performed the subtraction, you will see the term 2x plus 2. A-ha. It is now not possible to divide our x squared into 2x plus 2, because 2x plus 2 is only of degree 1. That means that we can’t go on, which is a sign that we should stop, and we do stop. And this 2x plus 2 is the remainder, capital R. The terms that we have carefully recorded above give us the quotient, Q, and the answer to our division then is the quotient plus the remainder over the divisor.

To summarise, we’ve divided P by D, and we’ve gotten the answer in standard form with the quotient and the remainder over the divisor. Multiplying across by D, we could write P as DQ plus R. When the remainder is 0, a 0 function, then we say that P is divisible by D. We’ll see that this algorithm for division of polynomials has highly useful consequences for calculating roots of polynomials in the next segment.