Algebraic Sum of Polynomials

In this lesson we will see the first two operations that can be performed on polynomials:

• The addition between polynomials
• The subtraction between polynomials

From an algebraic point of view, the two operations are essentially identical, since the subtraction between two polynomials consists of the sum between the first and the opposite polynomial of the second. For this reason, we speak, generally, of algebraic sum between polynomials.

We will see examples of algebraic sum between polynomials and extend everything to the case of algebraic sum of more than two polynomials.

Addition between two polynomials

Performing addition between two polynomials is quite simple. The result, in fact, consists of that polynomial composed of all the terms that make up the addend polynomials.

Let's try to examine an example. Suppose we want to add together the following polynomials and :

The sum operation is indicated in this way:

The result, i.e. the sum polynomial, is composed of all the terms of both and , so:

As you can see, the sum polynomial is not necessarily reduced to normal form. In this case, we can add all the similar terms together:

Subtraction between two polynomials

The subtraction between two polynomials can be essentially reduced to an addition. In fact, if we want to calculate the difference between two polynomials, in particular calculate the difference between polynomial and polynomial , it is sufficient to rewrite the subtraction operation in this way:

We changed the sign of and reduced the subtraction operation to an addition. In practice, however, changing the sign of a polynomial consists in replacing it with its opposite polynomial.

Opposite Polynomial

For example, given the polynomial:

Its opposite polynomial is obtained by changing the signs of all terms. Therefore, the result will be:

Example of subtraction between two polynomials

Having defined the concept of opposite polynomial, we can now examine an example of subtraction between two polynomials. Let's suppose we want to determine the difference between the following two polynomials, and :

The difference polynomial will be:

But, as we have seen, we can transform this operation into the sum between and the opposite of polynomial :

Therefore, we obtain the result:

The result contains similar terms that can be added together to reduce the polynomial to normal form:

Algebraic sum of several polynomials

We have seen that the operation of subtraction between two polynomials can be reduced to the operation of addition. For this reason, in general, we speak of Algebraic Addition of polynomials, without distinguishing between the two operations.

As such, the operation of Algebraic Addition can be extended, taking advantage of the associative property of addition, to the case of more than two polynomials. To better understand this, let's examine an example.

Let's suppose we have the following polynomials:

We want to determine the following algebraic sum of the three polynomials above:

To do this, it is sufficient to substitute the polynomials in the expression:

In the case of the last polynomial, that is polynomial , we replace it with its opposite polynomial:

From this point onwards, we can proceed as usual by adding together the similar monomials:

Summarizing

In this lesson, we learned that the operations of addition and subtraction between two polynomials essentially consists of the same operation of algebraic addition between polynomials.

In the case of addition, the sum of polynomials consists of a polynomial composed of all the monomials that make up the starting polynomials.

In the case of subtraction between polynomials, on the other hand, we add the first polynomial to the opposite of the second.

Obtaining the opposite polynomial of a given polynomial is very simple: we just need to reverse the signs of all the monomials that make it up.

Finally, we extended the operation of algebraic addition between polynomials to the case of addition between more than two polynomials.