Division of Polynomials

A polynomial is divisible by another polynomial when there exists a third polynomial, called the quotient, such that the first polynomial is equal to the product of the quotient and the second polynomial.

A necessary condition for a polynomial to be divisible by another polynomial is that the degree of the first one is greater than or equal to the degree of the second one. In fact, the degree of the quotient is equal to the difference between the degree of the dividend and the degree of the divisor.

In this lesson, we will see how to divide one polynomial by another. We will study how to obtain the quotient and the remainder of the division in cases where the dividend is not divisible by the divisor.

Divisibility Among Polynomials

To define the divisibility among polynomials, we start from the concept of divisibility among integers. We know that an integer number is divisible by another integer number when there exists a third number such that .

For example, the number 12 is divisible by 3. In fact, there exists a number, 4, which when multiplied by 3 yields 12.

In the same way, we can state:

Let's see a few examples:

Degree of the Quotient

From our lesson on polynomial multiplication, we know that the degree of the product of two polynomials is the sum of the degrees of the two polynomials.

Therefore, if a polynomial is divisible by another polynomial and knowing that is the quotient, we deduce:

Thus, the degree of the polynomial is equal to the sum of the degrees of the polynomials and :

From which we infer:

Therefore:

Let's revisit the previous example:

From this example, we can derive another important result:

Polynomial Division with Remainder

Given two polynomials and in the variable with the degree of greater than or equal to that of , we can always perform the division between the two polynomials to obtain a quotient and a remainder .

In fact, the following theorem exists:

In the special case where the polynomial is zero, then the polynomial is divisible by the polynomial and the polynomial division is an exact division.

Polynomial Division Procedure: Long Division

Now let's look in detail at how to perform polynomial division. This procedure is known as long division.

To understand how to proceed, let's start with an example.

Take the polynomial :

and the polynomial :

The first step is to check the degrees of the two polynomials. To do this, we must first ensure that and are in standard form. Indeed, the two polynomials above are already in standard form.

The degree of polynomial is and that of polynomial is . Therefore, the degree of the dividend polynomial is greater: the division can be performed.

The second step is to arrange the two polynomials in decreasing order of degree, also filling in with zeros the missing terms of the dividend. By doing this, we obtain:

In the example above, the dividend polynomial lacks the term of degree , so we fill it in with a zero. This is a fundamental step for polynomial division.

The next step involves dividing the two highest-degree terms by each other. In this case, the two highest-degree terms are:

Since they are two monomials, we use the monomial division. Thus:

We take this result and label it as : this is the first partial quotient. We write it above the dividend polynomial as follows:

At this stage, multiply the partial quotient by the divisor polynomial:

Then we put the result under the dividend polynomial, aligning the terms of the same degree:

The next step is to subtract the two polynomials:

The result is a polynomial that represents the new dividend.

At this point, we repeat the process to obtain the second partial quotient. Thus, we divide the highest degree terms of the new dividend and the divisor polynomial by each other:

We add this result to the previous partial quotient:

Then we write this result above the dividend polynomial:

Next, we multiply the partial quotient, in this case , by the divisor polynomial again:

We subtract this result from the new dividend:

In this case too, we have obtained a new dividend. The procedure continues until the degree of the new dividend is less than the degree of the divisor polynomial. In this instance, the degree of the new dividend is and that of the divisor polynomial is also . Therefore, the process can go on.

Thus, we divide the highest degree terms of the new dividend and the divisor polynomial by each other:

We add this result to the previous partial quotient:

Now, we multiply the partial quotient, in this case , by the divisor polynomial:

Then, we subtract this result from the new dividend:

The degree of the new dividend is and that of the divisor polynomial is . Therefore, the process cannot continue. We have completed the division.

In particular, the sum of the partial quotients is the quotient:

while the last dividend found represents the remainder:

Verification of the Result

We can verify the result of the division in a very simple way. Multiply the quotient by the divisor polynomial and add the result to the remainder. We expect the result to be equal to the dividend:

The result is equal to the dividend, so the division was performed correctly.

Short Division Method

There's an alternative method to perform the division between polynomials, called the short division method.

This method is even more compact compared to the method discussed above. However, short division requires some steps to be performed mentally.

Let's use the previous division between the two polynomials:

First, we list the dividend and divisor polynomials above each other in descending order of degree, also including the missing terms of the dividend:

As done above, we first divide the highest degree terms of the two polynomials: and . Thus:

We record this result below the line, aligning with terms of the same degree:

Since the division of the two terms, and , is exact, meaning it produced no remainder, we can cross out the term from the dividend as it has been used up:

Next, we multiply the result we just found by the remaining terms of the divisor:

Subtract this result from the dividend, specifically subtracting it from the term of the dividend of the corresponding degree:

The term of the dividend has been used up so we cross it out and report this result above it as a carry-over:

We repeat the procedure and divide the highest degree terms of the new dividend and the divisor between them: and .

This result is exact, in the sense that it did not produce a remainder. Therefore, we can cross out the term from the dividend and report the result below the line:

Multiply the result just obtained by the remaining terms of the divisor:

Subtract this result from the dividend, specifically subtracting it from the corresponding degree term of the dividend:

The term of the dividend has been used, so we cross it out and report this result above it as a carryover:

We repeat the process again by dividing the highest degree terms of the new dividend and the divisor: and .

In this case too, the division did not produce a remainder. Therefore, we can cross out the term from the dividend and report the result below the line:

Multiply the result we just obtained by the remaining terms of the divisor:

Subtract this result from the dividend, specifically subtracting it from the corresponding degree term of the dividend:

The term of the dividend has been used, so we cross it out and report this result above it as a carryover:

At this point, we can no longer continue because the degree of the new dividend is lower than the degree of the divisor polynomial. Therefore, we have concluded the division.

Specifically, the polynomial located below, under the line, is the quotient:

While the polynomial located above, over the line, is the remainder:

As you can see, the Short Division method is very compact but requires some mental calculations. Therefore, if you are new to polynomial division, it's recommended to use the notation seen earlier.

In Summary

Polynomial division is an operation that involves finding the quotient and the remainder of the division between two polynomials. The quotient polynomial is the result of the division, while the remainder polynomial is the remainder of the division.

The procedure to find the quotient and the remainder consists of the following steps:

1. Ensure that both the dividend and the divisor are in standard form. In particular, the degree of the dividend should be greater than or equal to the degree of the divisor polynomial.
2. Divide the highest-degree terms of the dividend and the divisor by each other. The result is the partial quotient.
3. Multiply the partial quotient by the divisor polynomial and subtract the result from the dividend. The result is the new dividend.
4. Repeat steps 2 and 3 until the degree of the new dividend is less than the degree of the divisor polynomial. The result is the sum of the partial quotients which is the quotient. The remainder is the last dividend found.

This process can be simplified in the case where the divisor is a polynomial of degree of the form . In this case, the Ruffini's Rule can be applied, as we will see in the next lesson.