Product between Monomials and Polynomials

The product between a monomial and a polynomial is an operation between polynomials that exploits the distributive property of multiplication.

The result is indeed a polynomial composed of the ordered sum of the products of the monomial by the terms of the starting polynomial.

In this lesson, we will see, through some examples, how to calculate the polynomial resulting from the multiplication between monomials and polynomials. We will also see how the degree of the resulting polynomial is related to the degrees of the monomial and the starting polynomial.

Product between a monomial and a polynomial

The operation of multiplication between a monomial and a polynomial is the basis of the operation of multiplication between polynomials.

To perform this operation, we need to apply the distributive property of multiplication. In fact, trying to solve the following expression:

we observe that the result will be:

In other words, the result is composed of the ordered sum of the products of by the individual addends , , and .

Bringing this process to the case of multiplication between a monomial and a polynomial, the result will be composed of the ordered sum of the products of the monomial by the individual terms of the polynomial.

Let's try to see an example:

By applying the distributive property of multiplication, we can calculate the result by rewriting the above expression as a sum of products of the monomial by the individual terms of the polynomial, paying particular attention to the signs:

At this point, it's a matter of applying the operation of multiplication between monomials several times, like this:

The resulting polynomial is, in this case, already in normal form.

So, to recap:

Degree of the resulting polynomial

It can be easily observed that the result of multiplying a monomial by a polynomial will have a total degree equal to the sum of the total degrees of the given monomial and polynomial.

Referring back to the previous example, we can see that the monomial has a total degree of 2. As for the given polynomial:

Its terms have the degrees listed in the table:

Therefore, the degree of the polynomial is 2, which is the maximum degree among its terms.

If we consider the result of the multiplication:

We can observe, as a first step, that this polynomial is already in standard form and is composed of the following terms:

We can see that the maximum degree of the terms in the resulting polynomial is 4, which also represents the degree of the polynomial and is equal to the sum of the degrees of the given monomial and polynomial: .

This result can be easily demonstrated. We can start from the property of the product of powers. In fact, when multiplying powers with the same base, we obtain a power with the same base and with an exponent equal to the sum of the exponents:

Applying this property to the case of the product of monomials, it is easy to see that the degree of the resulting monomial is equal to the sum of the degrees. Let's take an example:

Both monomials have a degree of 2. When multiplying them, we have to add the exponents of the variable :

Therefore, the result will have a degree of 4.

In the case of a product between a monomial and a polynomial, it is sufficient to iteratively apply this reasoning to the individual products between the monomial and the terms of the polynomial.

Further examples

Let's see some other examples.

Example 1

Let's try to calculate the result of the following multiplication:

Before performing the calculations, let's see what the overall degrees of the monomial and the polynomial are. The monomial has degree 4. The polynomial is already in normal form and is composed of the following terms:

Therefore, the polynomial has a total degree of 2. Therefore, the resulting polynomial will have a degree equal to .

By applying the distributive property of multiplication and paying attention to signs, we can rewrite the multiplication as:

Then, we calculate the individual products between monomials to obtain the final result:

The result is already in normal form and consists of the following terms:

Therefore, the result has a total degree of 6, as we anticipated above.

Example 2

Let's try to calculate the result of the following multiplication:

As in the previous example, we see what are the overall degrees of the monomial and the polynomial. The monomial has degree 2. The polynomial, on the other hand, is already in normal form and is composed of the following terms:

Therefore, the polynomial has an overall degree equal to 4. From this it follows that the resulting polynomial will have an overall degree equal to .

Let's now apply the distributive property of multiplication:

Then, we calculate the individual products of the monomials to obtain the final result:

The result, also already in normal form, is composed of the following terms:

Therefore, the result has a degree of 6.

Summary

We have seen in this lesson how it is possible to multiply a monomial by a polynomial.

The result will always be a polynomial that has as terms the products of the given monomial by the terms of the given polynomial. The overall degree of the result will be equal to the sum of the overall degrees of the starting monomial and polynomial.

The multiplication operation between a monomial and a polynomial is the basis of the multiplication operation between polynomials that we will see in the next lesson.