Product between Polynomials

Multiplication of polynomials is an operation that results in the ordered sum of all the products between the terms of the first polynomial and the terms of the second polynomial.

To calculate the product of polynomials, one must use the distributive property of multiplication. At the core of this operation is the calculation of the product between a monomial and a polynomial, as seen in the previous lesson.

In this lesson, we will go through the steps for calculating the product of two polynomials, study the degree of the result, and relate it to the degrees of the factored polynomials.

Furthermore, we will see how it is possible to interpret the product of polynomials geometrically and extend the operation to the case of the product of more than two polynomials.

Multiplication of Two Polynomials

The operation of multiplication of polynomials can be solved by applying the distributive property of multiplication. The difference compared to the case of the product between a monomial and a polynomial is that this property needs to be applied multiple times.

To better understand this, let's consider the following expression:

By applying the distributive property for the first time, we can distribute the factor to the terms and in this way:

Now, we apply the distributive property again, distributing both and to the terms of the sum , obtaining the final result:

In essence, the result is composed of the ordered sum of the products of with the terms and and the products of with the terms and .

Therefore, multiplying two polynomials consists in obtaining a polynomial composed of the ordered sum of the products of each term of the first polynomial with each term of the second polynomial:

Let's try to clarify this with an example. Suppose we want to multiply the following polynomials:

First, we apply the distributive property of multiplication, distributing the second factor to the terms of the first polynomial, in this way:

Subsequently, we apply the distributive property of multiplication individually to the two terms obtained:

The result obtained is a polynomial not in normal form, so we add the similar terms:

Degree of the resulting polynomial

Similarly to the case of a product between a monomial and a polynomial, in this case, it can be easily shown that the degree of the product between two polynomials is equal to the sum of their degrees.

Returning to the previous example, we see that the first polynomial is composed of the following terms:

Therefore, the overall degree of the first polynomial factor is 1.

The second polynomial factor is composed of the following terms:

The degree of the second polynomial factor is 1.

Taking the result of their multiplication in normal form, we have that it is composed of the following terms:

In this case, the maximum degree is 2 and it is precisely equal to the sum of the degrees of the two polynomial factors.

This result is easily demonstrable using the property of the product of powers. By multiplying powers with the same base, we obtain as a result a power with the same base but with the exponent equal to the sum of the exponents:

From this, it is easy to see how the degree of the product of two monomials is equal to the sum of the degrees of the factor monomials. For example:

In this example, the two factor monomials have degree equal to 3, while the result has degree equal to 6.

Given that the product of two polynomials is obtained by iteratively calculating the product between monomials, and given that the degree of the two polynomials is given by the maximum degree of the monomials that compose them, it is easy to understand how the resulting product polynomial must have a degree equal to the sum of the degrees of the factor polynomials.

Geometric Interpretation

An interesting geometric interpretation can be given to the product of polynomials.

Suppose, in fact, we want to calculate the area of a rectangle with base and height . Suppose also that both the base and the height are given by the sum of two segments, respectively:

The base is given by the sum of the segments and , while the height by the sum of the segments and , as shown in the following figure:

To calculate the area of the rectangle we need to multiply the base by the height, therefore:

So the area of the rectangle is equal to the product of two polynomials. If we expand the product of the two polynomials we get the result:

In other words, the area of the rectangle is given by the sum of 4 terms. By observing the figure below we can see that the 4 terms correspond to the areas of the sub-rectangles whose union represents the main rectangle, as shown in the following figure:

Further examples

Let's see some other examples.

Example 1

Let's try to compute the following product of two polynomials:

First of all, distribute the second polynomial factor over the two terms of the first polynomial:

By applying the distributive property of multiplication, distribute the two factors and over the terms of the second polynomial:

Then, compute the products between monomials that make up the sum:

Finally, put the polynomial in standard form by adding together the similar terms:

The final result has degree 4, since it is composed of the following terms:

This confirms the degree property of a product polynomial, since both factor polynomials have a degree of 2, so the degree of the result is :

Example 2

Let's consider the following product of two polynomials:

Also in this case, we apply the distributive property of multiplication by distributing the second polynomial factor to both terms of the first polynomial:

Then, we distribute the factors and to the terms of the second polynomial:

Next, we calculate all the monomial products that compose the sum:

Finally, we add together the similar monomials to bring the resulting polynomial to standard form:

The resulting polynomial is composed of the following terms:

As can be observed, the degree of the final product is 3, which is equal to the sum of the degrees of the two starting factor polynomials:

Extension to the case of more than two polynomials

We can extend the operation of polynomial multiplication to the case in which we have more than two polynomials. The procedure is quite simple, we just need to perform the multiplications in order.

To clarify this, let's look at the following example:

In this case, we are dealing with three polynomials. To solve this multiplication, we label the three polynomials as , , and :

We can then rewrite the multiplication as:

We can, therefore, use the associative property of multiplication to first calculate the product of , and then calculate the product of the result with , as follows:

First, let's calculate using the procedure seen above:

This polynomial is already in normal form as it does not contain similar terms.

At this point, we multiply the previously obtained result by the polynomial :

Using the same procedure as seen above for the case of the product of two polynomials, we obtain:

The final polynomial is already in normal form and is composed of the following terms:

Therefore, the degree of the final polynomial is 7.

Observing the starting factor polynomials, we see that:

This confirms the property of the degree of the product polynomial even for the case of more than two polynomials. In fact, the degree of the resulting final polynomial is equal to , which is the sum of the degrees of the factor polynomials.

Summary

In this lesson, we have seen how to calculate the product of two polynomials. It involves applying the distributive property of multiplication multiple times. The result is the sum of each term of the first polynomial multiplied by each term of the second polynomial.

We have also seen that the degree of the resulting product polynomial is equal to the sum of the degrees of the factor polynomials.

Finally, we have extended the polynomial multiplication operation to the case of more than two polynomials by exploiting the associative property of multiplication.

The procedure seen in this lesson deals with computing the general case of polynomial multiplication. There are special cases of polynomial products that are known as Algebraic Identities. In such cases, it is not necessary to apply the procedure seen, as the result can be obtained almost immediately.

In the next lesson, we will see what are the most important Algebraic Identities.