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 $a$ is divisible by another integer $b$ if there exists a third number $c$ such that when multiplied by $b$ , it results in $a$ :

a \in \mathbb{Z} \quad \text{is divisible by} \quad b \in \mathbb{Z} \quad \Leftrightarrow \quad \exists \, c \in \mathbb{Z} : b \cdot c = a

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

3 \cdot 4 = 12 \quad \Rightarrow \quad 12 \quad \text{is divisible by} \quad 3

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:

2a + a^2b, \quad 2a

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

1 + \frac{1}{2}ab

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

\left( 1 + \frac{1}{2}ab \right) \cdot 2a \quad =

= \quad 2a + \left( \frac{1}{\cancel{2}}ab \right) \cdot \cancel{2}a =

= \quad 2a + a^2b

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

P \text{ is divisible by } m \quad \Leftrightarrow \quad \exists \, Q : Q \cdot m = P

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:

P: \quad 8x^7 - 2x^6 + 10x^4

and the monomial:

m: \quad 2x^3

We ask ourselves if $P$ is divisible by $m$ .

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

\frac{8x^7 - 2x^6 + 10x^4}{2x^3}

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

\frac{8x^7}{2x^3} - \frac{2x^6}{2x^3} + \frac{10x^4}{2x^3}

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

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:

4x^4 -x^3 + 5x

From the above example, we can draw two conclusions:

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;
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 $P$ is divisible by a monomial $m$ if and only if:

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

Examples

Let's see some examples:

Example

Consider the polynomial:

P: \quad 3ab + 5a^2c

and the monomial:

m: \quad 2ac

In this case, the polynomial $P$ is not divisible by $m$ .

In fact, the first term of the polynomial, $3ab$ , does not contain the letter $c$ from the monomial.

Example

Consider the polynomial:

P: \quad 3a^2b + 5a

and the monomial:

m: \quad a^2

In this case, the polynomial $P$ is not divisible by $m$ .

In fact, the second term of the polynomial, $5a$ , has the exponent of the letter $a$ equal to $1$ . However, in the monomial, the letter $a$ has an exponent equal to $2$ .

Example

Consider the polynomial:

P: \quad 4xy^4 - x^2y^2

and the monomial:

m: \quad 3xy^2

In this case, the polynomial $P$ is divisible by $m$ .

Dividing $P$ by $m$ we get:

\frac{4xy^4 - x^2y^2}{3xy^2}

= \quad \frac{4xy^4}{3xy^2} - \frac{x^2y^2}{3xy^2}

= \quad \frac{4}{3}y^2 - \frac{1}{3}x

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:

P: \quad 20x^4 - 12x^3 + 6x^2

and the monomial:

m: \quad 2x^2

The polynomial $P$ has a degree of $4$ . The monomial $m$ has a degree of $2$ .

According to the rule above, the degree of the quotient polynomial will be $4-2 = 2$ . By performing the division, we can verify that this result is correct:

\frac{20x^4 - 12x^3 + 6x^2}{2x^2}

= \quad \frac{20x^4}{2x^2} - \frac{12x^3}{2x^2} + \frac{6x^2}{2x^2}

= \quad 10x^2 - 6x + 3

The quotient polynomial has a degree of $2$ , 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.