Operations with Monomials

The operation with monomials is the basis of literal calculations.

In this lesson we will look at the main operations between monomials that can be performed:

• Addition of monomials, the result of which is called the sum,
• Difference of monomials, the result of which is called difference and can be seen as a special case of algebraic addition of monomials,
• Multiplication of Monomials, the result of which is called the product,
• Power of a monomial,
• Division of monomials, the result of which is called the quotient.

In particular, we are interested in understanding whether the result of an operation between monomials always yields a monomial.

Algebraic addition of monomials

The first operation with monomials that we will examine is the addition of monomials.

Before seeing how the sum between monomials can be calculated, let's start with some examples. Consider the following addition:

Let's try for a moment to forget that we are dealing with two monomials. Just consider the whole thing as the addition of two expressions consisting only of multiplications. We can see that there are common factors in the two addends. In particular, the common factors are and . Therefore, we can collect as a common factor the expression :

The end result is a monomial.

Conversely, consider the following example:

In this case, we cannot simplify the addition as we did before. There are no common factors that allow us to simplify the expression into a monomial. This does not at all imply that the expression is invalid. The issue is that the expression cannot be reduced to a monomial.

If we look closely, the difference between the two additions is that in the first case we had an addition between similar monomials, i.e. with the same literal part. In the second case, however, the monomials were not similar to each other.

From these two examples, we can see that the algebraic sum of two monomials is a monomial if and only if the two addends are similar monomials. In other words, the algebraic sum of monomials can only be defined for similar monomials.

Let's look at another example:

Instead, let's try to add two opposite monomials:

Thus the algebraic sum of a monomial and its opposite is equal to zero.

In the same way as relative numbers, the difference of two monomials can be seen as the result of the algebraic addition between the first monomial and the opposite monomial of the second. Hence:

For example:

Like relative numbers, the operations of addition and subtraction for monomials can be considered as the same operation from an algebraic point of view. Therefore, in short, addition and subtraction of monomials are referred to as algebraic addition of monomials and the result as algebraic addition of monomials.

Multiplication of monomials

To study the multiplication of monomials, we start with an example. Let us consider the multiplication between two monomials that follows:

First we apply the commutative property of multiplication to group numerical factors and literal powers with equal bases:

Then we use the associative property of multiplication to multiply numerical factors and literal powers with equal bases:

Finally, we apply the first property of powers:

The end result, the product of two monomials, is itself a monomial.

In the example, although the two monomials are not similar, the result is always a monomial. This is why, unlike addition, the product of two monomials is always a monomial. We can now define the multiplication of monomials:

Power of a monomial

The power of a monomial is a fairly simple operation. Let us try to examine an example:

To obtain the result, it is sufficient to exploit two properties of powers. The first is the property of the power of a product: . Hence we get:

Next, we must apply the property of powers of powers: . So the expression becomes:

So the end result is:

As we can see, the result of a power of a monomial is itself a monomial. From this we can derive the general rule:

Using the properties of powers, we can also calculate the case of a monomial raised to 0. In this case, any monomial raised to 0 is equal to 1. This applies as long as the monomial is different from 0, i.e. from null monomial. For example:

Similarly, any monomial raised to 1 is equal to the monomial itself:

Division of monomials

The operation of dividing monomials is more complex than the others because the result, the quotient, is not necessarily a monomial.

To understand this, let's look at two examples. Let's take a first example:

In order to solve this division, we can exploit two properties:

1. The invariant property of division:
2. The commutative property of multiplication.

In this way we can separate the coefficients from the literal parts:

Note that in the monomial divisor, we have added a new literal factor: . This factor is equal to 1, so its addition is legal. In this manner, we have the same letters for both the dividend and the divisor, which simplifies the calculations.

Now we can apply the property of division between powers: . Hence:

The result is a valid monomial.

Now, using the same procedure as above, we try to solve the following division:

The result we have obtained is not a monomial. In fact, the letter has a negative exponent.

So division between monomials does not always yield a monomial. Note that this does not mean that the resulting expression is meaningless, on the contrary, from a mathematical point of view it makes perfect sense.

The concept of divisibility must be introduced.

Divisibility between monomials

The divisor monomial cannot be the null monomial, i.e. zero. In fact, such a writing has no meaning:

We can now generalise the process of division between monomials:

It follows from the definition of divisibility and quotient that every monomial is divisible by any number. That is, it is possible to divide a monomial by any number.

The quotient of a monomial divided by a number is simply the monomial in which the literal part is the same and the coefficient is equal to the coefficient of the monomial divided by the number in object. For example:

In brief

In this lesson, we have seen the various operations that can be carried out on monomials:

• Algebraic addition of monomials.
• Multiplication of monomials.
• Power of a monomial
• Division of monomials

In particular, we focused on cases in which the result of these operations is itself a monomial. For the case of addition, this means when the monomials are similar, and for division when the dividend is divisible by the divisor.

In the next lesson we will use the operations we have just seen to calculate the Lowest Common Multiple (lcm) and Greatest Common Divisor (GCD) between monomials.