Polynomial Division

Now I’ll explain the division of polynomials. I will teach you how to divide a polynomial into another polynomial with the general method, with step-by-step exercises.

The division of polynomials is one of the topics that generates the most doubts, because it is necessary to take into account many concepts that are not normally fully understood and this causes many mistakes to be made.

Properties of the polynomial division

Before we go on, so that you can better understand this method, let’s remember the parts that make up any division, be it a division between numbers or between polynomials.

Any division is made up of the dividend, the divisor, the quotient and the remainder:

how to split polynomials

If we write these parts in the form of a polynomial, it remains

polynomial between polynomial

And they fulfill the following properties:

1- The grade of the dividend D(x) is greater than or equal to the grade of the divisor d(x):

methods to divide polynomials

2- The grade of the dividend D(x) is equal to the grade of the divisor d(x) plus the grade of the quotient C(x):

how to divide polynomials by polynomials

3- The grade of the remainder R(x) is less than the grade of the divisor d(x):

how to divide polynomials between polynomials

4- The dividend is equal to the divider by the quotient plus the rest:

polynomials division

If you check this formula with a division of numbers, you will see that it is also met.

Polynomial division

The method I am going to explain to you corresponds to the general method of division of polynomials, since there are more methods to divide polynomials, such as the Ruffini method or rule, which we will see later.

This method is used to divide any type of polynomials and to do it you have to take into account the previous properties.

Let’s look at it with a step-by-step example:

division of polynomials

The first step is to correctly place and write down the dividend and divider in order to begin your division.

In our case, the numerator is the dividend and the denominator is the divisor.

Both the dividend and the divisor are written in descending order of the degrees of their terms, i.e. starting with the highest grade, and ending with the end of grade 0 (the independent term).

In addition, if the term of any degree is missing in the dividing line, a space is left in its place.

In our example, the dividend does not have a grade 2 term, so we leave a space in its place. It looks like this:

divisions with polynomials

Once we have the dividend and the divisor in place, and with the corresponding gaps in the missing terms, we will begin to calculate the quotient.

To do this, we divide the first term of the dividend by the first term of the divisor:

divide polynomial by polynomial

divide a polynomial

And we put it in the quotient. Corresponds to the first term of the quotient:

how to divide a polynomial between another polynomial

Now, we must multiply this quotient term by each of the divisor terms:

divisions between polynomials

We put them under the dividend, but in compliance with these two conditions:

  • Each one is placed below its similar term, i.e., the term has the same grade or its corresponding gap (in the event that the dividend has no grade term)
  • With the opposite sign

It looks like this:

how to divide polynomial by polynomial

As we left a hole for the term grade 2, we placed the 6x² underneath that hole.

And now for the dividend, we add vertically the two expressions we have:

divisiones polinomios

By having each term below its similar dividend term, this sum is realized in a more orderly manner. This is also what we are looking for when we leave the gap in the dividend of the missing term.

In doing so, the term of highest degree is annulled, which is the goal of all the steps we have taken so far.

At this point, we have a new algebraic expression left in the dividend whose degree is greater than the degree of the divisor.

You have to keep dividing until the expression left on the dividend is less than the divisor.

Therefore, we continue to divide this new expression between the divisor, repeating all the steps again:

To calculate the second term of the quotient, we divide the first term of the new dividend expression by the first term of the divisor:

properties of polynomial division

how to split polynomials

And we put it in the quotient. It corresponds to the second term of the quotient:

divisions of polynomials

Now, as before, we multiply this second term of the quotient by each of the divisor terms and place them in the part of the dividend, below its similar term and with the opposite sign:

division of a polynomial into a polynomial

We add vertically in the dividend:

division of a polynomial into another polynomial

By nullifying the term of highest degree and obtaining a new expression, whose degree is now equal to the degree of the divisor, so we can still continue to divide.

We repeat the process again to calculate the third term of the quotient. We divide the first term of the new dividend expression by the first term of the divisor:

division of polynomial by polynomial

polynomial divisions between polynomials

And we place it in the quotient, which corresponds to the third term of the quotient:

division de polinomios

We continue to multiply this third term of the quotient by each of the divisor terms and place them in the part of the dividend, below its similar term and with the opposite sign:

division polinomios

And then we add vertically again:

division of polynomial between polynomial

Now, the resulting dividend expression is less than the divisor grade. Therefore, we can no longer continue, so we have finished dividing.

The last expression left in the dividend part, with a degree lower than the degree of the divisor, corresponds to the rest of R(x) and the expression we have calculated below the divisor corresponds to the quotient C(x).

Therefore, the result of the division is the quotient C(x):

division of polynomials between polynomials

And the rest R(x):

divide polynomials

If we want to express the dividend D(x) according to the divisor d(x), the quotient C(x) and the rest R(x), we can do it following the fourth property of the polynomial division:

steps to divide polynomials

These would be all the polynomials we need:

división polinomio entre polinomio

That if we replace it, it’s left to us:

division de polinomios