If we focus on colloquial language, we could say that **remarkable products** are those **goods** that can be purchased in the market and that have special characteristics: a luxury car, a gold watch, a last generation computer ...

The notion of **remarkable products** However, it does not usually refer to this issue, but is used in the **math** to name certain **algebraic expressions** that can **factor immediately** , without resorting to a process of various steps.

In this regard, we must remember that the concept of **product** , in the mathematical field, refers to the result of a **multiplication operation** . The values that come into play in these operations, on the other hand, are known as **factors** .

An algebraic expression that appears frequently and can undergo a factorization with the naked eye, therefore, is called a remarkable product. A **square binomial** and the **product of two conjugated binomials** are **examples** of notable products.

A concrete example of binomial squared is as follows:

**(m + n) ² = m² + 2mn + n²**

Said notable product refers to the square of the sum of **m** and **n** is equal to the square of **m** more twice **m** multiplied by **n** plus the square of **n** .

We can check it by replacing the terms with **values** numeric:

(2 + 4) ² = 2² + 2 x 2 x 4 + 4²

6²= 4 + 16 + 16

36 = 36

In this way, if we find the square of a binomial as in the previous example, we can factor it immediately, without having to resort to all the steps, since it is a **remarkable product** .

The binomial squared can also consist of the subtraction of the two variables that are squared. In this case, the difference with respect to the previous example is that to solve it, the first one must be reversed. **plus sign** after the **same** , so that the following is **equation** :

**(m - n) ² = m² - 2mn + n²**

In addition to the squared binomial, notable products are divided into the following types (the equations can be seen in the image):

*** Binomial sum by binomial difference** : it is the product between a binomial in which its variables are added and another, in which they are subtracted. To solve it, simply subtract the **square** of each variable;

*** Binomial to the cube** : as well as the squared binomial, it is also divided into addition and subtraction. In the first case, it is the cube of the sum of two variables, which is equal to the square of the first plus three times the first squared by the second, plus three times the first by the second squared, plus the second cubed . For subtraction, the first and last must be reversed **plus sign** ;

*** Sum of cubes** : when the product is observed between the sum of two variables, and the first squared minus the first by the second plus the second squared, there is a very simple way to solve it, which consists of adding the cube of the first **variable** to the second.

With respect to the applications of notable products, it goes without saying that they are not found in the daily life of most people, as it is perhaps the case with the simple rule of three, for example, among others of the most accessible topics of mathematics. However, professionals from various sectors take advantage of notable products; Let's look at three examples below:

***** the **engineers** civilians use it to measure distances, volumes and areas;***** it is used to calculate the intensity of the electric current;***** it allows to carry out an estimate of the number of individuals that are in a genetic algorithm;***** serves to calculate the torsion of various **structures** .