Division of a Polynomial by a Monomial

At the heart of polynomial division is the operation of dividing a polynomial by a monomial. Dividing a polynomial by a monomial means dividing all the terms of the polynomial by the monomial.

From the division of polynomials and monomials, we can derive the concept of divisibility of a polynomial by a monomial. In this lesson, we will see how to verify whether a polynomial is divisible, or not, by a monomial.

Using the properties of divisions, we will see how to operationally perform a division between a polynomial and a monomial.

Key Takeaways
  • Dividing a polynomial by a monomial involves dividing each individual term of the polynomial by that monomial;
  • A polynomial is divisible by a monomial if there exists a quotient polynomial that, when multiplied by the monomial, gives the original polynomial;
  • The degree of the quotient polynomial is equal to the degree of the polynomial minus the degree of the monomial.

Dividing a Polynomial by a Monomial

The fundamental prerequisite for understanding polynomial division is knowing how to divide a polynomial by a monomial.

However, before diving into that, we need to understand when a polynomial is divisible by a monomial. Let's start by looking at the divisibility of integers. We can say that an integer is divisible by another integer if there exists a third number such that when multiplied by , it results in :

For example, the number 12 is divisible by 3 because there exists a number, 4, that when multiplied by 3 yields 12:

Obviously, no integer is divisible by zero.

We can use the same reasoning for polynomials and monomials. Thus, a polynomial is divisible by a monomial if there exists a polynomial that when multiplied by the monomial yields the original polynomial.

Example

Consider the following polynomial and monomial:

The given polynomial is divisible by the monomial. In fact, there exists the following polynomial:

If we multiply this polynomial by the monomial, we get:

This results in the original polynomial.

So, we can provide a definition for divisibility of a polynomial by a monomial as follows:

Definition

Divisibility of a Polynomial by a Monomial

A polynomial is divisible by a monomial (where is non-zero) if there exists another polynomial, called the quotient polynomial, such that when multiplied by the monomial , it yields the polynomial :

Although this definition is mathematically sound, it's not very operationally helpful. It doesn't guide us on how to actually perform the division between a polynomial and a monomial.

Let's try to find a practical method by reasoning through an example.

Example

Consider the polynomial:

and the monomial:

We ask ourselves if is divisible by .

To answer this question, let's try dividing by :

But, using the properties of division, we can rewrite the expression as:

In other words, we have transformed the division of polynomial by monomial into a sum of divisions between monomials. Specifically, we have converted the expression into the sum of the terms of polynomial each divided by the monomial .

At this point, the rest is simple. In fact, we can leverage the division between monomials that we already covered in the last lesson. Thus, the result becomes:

From the above example, we can draw two conclusions:

  1. The division of a polynomial by a monomial is obtained by summing up the divisions of all the individual terms of the polynomial by the monomial;
  2. A polynomial is divisible by a monomial if all its terms are divisible by the said monomial.

Therefore:

Definition

Divisibility Criterion of a polynomial by a monomial

A polynomial is divisible by a monomial if all the monomials that compose it are divisible by that monomial.

In other words, a polynomial is divisible by a monomial if and only if:

  1. All the letters present in the monomial appear in the monomials that make up ;
  2. The exponents of the letters of the monomials that make up are greater than or equal to the exponents of the corresponding letters of the monomial .

Examples

Let's see some examples:

Example

Consider the polynomial:

and the monomial:

In this case, the polynomial is not divisible by .

In fact, the first term of the polynomial, , does not contain the letter from the monomial.

Example

Consider the polynomial:

and the monomial:

In this case, the polynomial is not divisible by .

In fact, the second term of the polynomial, , has the exponent of the letter equal to . However, in the monomial, the letter has an exponent equal to .

Example

Consider the polynomial:

and the monomial:

In this case, the polynomial is divisible by .

Dividing by we get:

Degree of the Quotient Polynomial

The result of dividing a polynomial by a monomial is always another polynomial, known as the quotient polynomial:

Definition

Quotient Polynomial

The result of dividing a polynomial by a monomial, given that the polynomial is divisible by the monomial, is always a polynomial termed the Quotient Polynomial.

Based on the divisibility criterion we discussed earlier, we can infer that if a polynomial is divisible by a monomial, the degree of the monomial divisor is less than or equal to the degree of the polynomial dividend. Hence, we can derive the following property of the quotient polynomial:

Definition

Degree of the Quotient Polynomial

The quotient polynomial resulting from the division of a polynomial by a monomial has a degree equal to the difference between the degree of the dividend polynomial and the degree of the divisor monomial.

Let's illustrate with an example:

Example

Consider the polynomial:

and the monomial:

The polynomial has a degree of . The monomial has a degree of .

According to the rule above, the degree of the quotient polynomial will be . By performing the division, we can verify that this result is correct:

The quotient polynomial has a degree of , and our earlier result is confirmed.

In Summary

In this lesson, we introduced a fundamental operation that is preparatory to the study of polynomial division: the division of a polynomial by a monomial.

Dividing a polynomial by a monomial involves dividing all the terms of the polynomial, which are monomials themselves, by the given monomial. We also derived a divisibility criterion that allows us to determine whether a polynomial is divisible by a monomial or not.

Lastly, we derived a relationship between the degree of the quotient polynomial and the degrees of the dividend polynomial and the divisor monomial.

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