Long Division with Integers

Suppose you are given positive integers $D$ and $d$. We want to find integers $q$ and $r$ such that $0 \leq r < d$ and $D = qd+r.$ This we can do with long division, which we all learned to do in elementary school. 

To refresh our memory, we divide $\frac{745}{13}$ by hand. We write the numbers down like this:

$\underline{5 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 13|745$

As $7$ is smaller than $13$, we group the first two digits together and we see that: $5\cdot 13 = 65 \leq 74 < 78 = 6\cdot 13 $

So we write down a five as our first digit of $q$ and subtract $65$ from $74:$

$\underline{ \ 5 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \begin{aligned}13|&745 \\ -&65 \\ &\ \ 95 \end{aligned}$

We now group the remaining two digits and see that $7 \cdot 13 = 91 \leq 95 < 104 = 8 \cdot 13$

So the second digit of $q$ is $8$ and we subtract $91$ from $95$ to obtain $4$. As $4$ is smaller than $13$, we cannot repeat this procedure and we have found that $q = 58, r = 4.$ So $745$ contains $58$ copies of $13$, and another copy of $4.$

$\underline{ \ 58 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \begin{aligned}13|&745 \\ -&65 \\ &\ \ 95 \\ - & \ \ 91 \\ & \ \ \ \ 4 \end{aligned}$

Dividing Polynomials with Long Division

The beauty of long division is that the algorithm can be used not for integers only, but also for polynomials. 

Here we think about a larger polynomial as one with a higher degree. So given two polynomials $D(x)$ (the dividend) and $d(x)$ (the divisor), we are looking for two polynomials $q(x)$ (the quotient) and $r(x)$ (the $remainder)$ such that $D(x) = d(x)q(x) + r(x)$ and the degree of $r(x)$ is strictly smaller than the degree of $d(x).$

Conceptually, we want to see how many copies of $d(x)$ are contained in $D(x)$ (this is the quotient) and then how far $D(x)$ is away from being a multiple of $d(x)$ (this is the remainder).

 For example, suppose we want to divide $6x^3-8x^2+4x-2$ by $2x-4$:

$\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 2x-4|6x^3-8x^2+4x-2$

We look at the highest degree terms and we see that $6x^3=2x\cdot3x^2$. So we write down a $3x^2$, multiply the divisor with this result and subtract this from the dividend:

$\underline{\ \ \ \ \ \ \ 3x^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 2x-4|6x^3-8x^2+4x-2 \\ \ \ \ \ \ - \ \ 6x^3 -12x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4x^2+4x-2$

Again looking at the highest degree terms, we see that $4x^2 = 2x\cdot2x$, so we write down $2x$ as the second term in the quotient and proceed as before:

$\underline{\ \ \ \ \ \ \ 3x^2 +2x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 2x-4|6x^3-8x^2+4x-2 \\ \ \ \ \ \ - \ \ 6x^3 -12x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4x^2+4x-2 \\ \ \ \ \ \ - \ \ \ \ \ \ \ \ \ \ \ \ \ 4x^2 - 8x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 12x-2$

As $12x = 2x\cdot 6$, our next term will be $6:$

$\underline{\ \ \ \ \ \ \ 3x^2 +2x +6 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 2x-4|6x^3-8x^2+4x-2 \\ \ \ \ \ \ - \ \ 6x^3 -12x^2 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4x^2+4x-2 \\ \ \ \ \ \ - \ \ \ \ \ \ \ \ \ \ \ \ \ 4x^2 - 8x \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 12x-2 \\ \ \ \ \ \ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 12x - 24 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 22$

As multiplying any polynomial with the divisor $2x-4$ gets us a polynomial of degree greater than $0$, we cannot divide any further. We see that the quotient $q(x)$ $3x^2+2x+6$ and the remainder $r(x)$ is $22$, so

$6x^3-8x^2+4x-2 = (2x-4)(3x^2+2x+6) + 22$. 

Zero Remainders and Factors

If the remainder $r(x)$ equals $0$, we also say that there is no remainder and do not explicitly write out the $0$. This means that $D(x)=d(x)q(x)$: the dividend is a multiple of the divisor, or the divisor is said to $$divide the dividend. We say that the divisor is a factor of the dividend. (Of course, the quotient will also be a factor.)

$$Checking Your Results

If you have enough time to check your results, it is always wise to do so. The best way to do this is to explicitly work out the equation 

$D(x) = d(x)q(x)+r(x)$.

Another way is to check this equation for only one value of $x$.