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Long Division of Polynomials

Just as we can divide two numbers using long division, we can also use the same process to divide two polynomials.  The easiest way to understand this process is to work through an example.

Let’s begin with a short review of how long division of numbers is done so that we can see the parallels when dividing polynomials.

Example 1

Compute this quotient:

Solution 1

We set this up as a long division problem and use the standard algorithm:

We first ask ourselves how many times 2 goes into 3.  The answer is 1 ten, so we place a 1 on top of the division,  multiply 1 by 2 and carry out subtraction to generate a remainder.

Then we ask ourselves how many times 2 goes into 15.  The answer is 7, so we multiply 7 by 2 and calculate the remainder:

Since 2 cannot go into 1, we must stop here and our solution is the remainder, 1, divided by the original denominator, 2, added to the expression on the top.  That is,

Long division of polynomials follows this same process.

Example 2

Compute this quotient:

.

Solution 2

We set this up in the same manner as long division of numbers,

Then we ask ourselves how many times goes into .  Since we would have to multiply x by x to get , we can conclude that goes into x-times with some remainder.  So, we place an x on top of the division and proceed in the same way as we would in division of numbers.

Now we ask ourselves how many times goes into .  The answer is -1 times.  So we add a -1 to the expression above the division and continue.

We have now reached a point at which will no longer go into the remainder.  Just as in long division of numbers the remainder, 1, divided by the original denominator, , is added to the expression above the division and we have the following:

In this case, we had a non-zero remainder, which means (as in division of numbers) that does not divide .

Let’s consider a case in which the remainder is zero.  You may carry out the long
division to confirm this, but it is true that

Note that there is no remainder term like we had in the previous example.  Multiplying both sides by , we see that .  Also note that if we substitute 3 in for x on the right hand side, we get zero.   This means that 3 is a root of the polynomial on the left hand side of the equation.  In other words, if divides a polynomial , then is a root of  (see the tutorial entitled Rational Root Test for a definition of root).