Polynomials are one of the most basic expressions that can be found in algebra, a completely fascinating world for all those who love mathematics. In fact, this is the first litmus test for all those who want to take their first steps in equations, one of the mandatory expressions in Spanish schools. Above, polynomials are an initial step before getting into equations of different degrees.
Basically, a polynomial becomes an equation since it has unknowns and allows infinite values to be represented within a completely finite line. Therefore, it is a mathematical expression that allows expressing different variables. Thus, solving polynomials is an obligation to take into account, in addition to knowing Ruffini’s rule.
What are polynomials?
As we have said before, a polynomial is a type of algebraic expression that is part of one of the most basic that can be found within the world of equations. Among these, there are between first and third degree, in addition to being able to count on the possibility of having different unknowns, generally represented by the letters x, y or z, among others. The two main elements of polynomials are the constants (they are permanent numbers that are also known as coefficients), and the variables themselves, which are the different options that can be obtained depending on the number that we assign to the constant. In this sense, in polynomials two main types of variables can be found.
On the one hand, the independent variable is the constant since it does not depend on any other extension to be able to work. Instead, as we have said before, the variable depends directly on the constant. If you don’t assign any type of value to the constant you can’t get any type of variable. Therefore, it is known as the dependent variable since it depends directly on the other variable. Any type of equation usually has a dependent variable and an independent variable. In fact, when representing it on the line, it is necessary to differentiate each one since it will depend on whether it is represented on the coordinate or abscissa axis.
Beyond the structure, polynomials continue to play a really important role today, since they have multiple uses in complex fields such as physics, chemistry, or even more social areas such as economics. When making a possible approximation of any type of function that can be derived, these types of expressions are some of the most used. The history of polynomials takes place in Ancient Greece for the most part, and many mathematical expressions that were developed later had their origins in polynomials. The best known case is that of notable identities.
The notable identities, derived from the polynomials
Notable identities are the best-known pairings today. They are polynomials of two variables expressed as a power and that are used to establish equivalences between two expressions that facilitate the addition of polynomials. In this way (x+y) squared is always expressed as the square of x + twice xy and the square of y. Thus, x+y squared may sound Chinese, but expressed in another way it may be easier to perform the polynomial operation. Remember that the number of letters is used to establish the number of variables that the polynomial or monomial in question consists of.
The coefficient corresponds to the number that multiplies the variables, while the exponent(s) are the numbers that appear above the variable, although in this case it is usually invisible. When no exponent appears, this will always be one. Don’t forget that any number raised to the power of one will always have the same coefficient. In addition, it also serves to establish the degree of any type of equation.
In general terms, there are three types of notable identities, although for the sum of polynomials we only find the expression already described, although do not forget that the exponent can vary and be higher than squared, and can be cubed or in more degrees. Do not forget that the higher the exponent, the greater the decomposition of products that we must carry out.
Benefits of knowing how to add polynomials
However, there are numerous benefits that we can get from knowing how to add polynomials in our daily lives. Some of the most prominent are the following:
- Pass math. Polynomials are an essential requirement if you want to pass the subject of mathematics. Part of the academic program involves passing this part of mathematics, and it will also serve to continue growing in this world since they are the basis of the equations.
- Establish relationships with derived functions. When it comes to crossing data, polynomials play a fundamental role.
- Get started in the world of equations. As you have read above, equations are a really complex world. The first step is in the polynomials. Therefore, it is a good way to test if they really are what you are looking for or instead it is not your thing.
- Develop a professional career. Many fields of vital importance such as physics, chemistry or economics use proper functions of polynomials. Thus, their learning and mastery are really necessary to continue progressing in the mathematical world.
How to add polynomials?
Once the main properties of polynomials are known, it is time to learn about the different methods we have to add polynomials and operate with them correctly. There are different methods and the first one we are going to learn is the horizontal one. To do this, we will use an example that will help us to illustrate the procedure. We start from the base that we want to add (x4 -3×2 + x + 1) and the other polynomial is (x3 -x2 + 5x -2). In this sense, from this procedure we will place them one next to the other horizontally. In this case, it is extremely important that we look at all the terms that have the same exponent and, therefore, are of the same degree.
With polynomials it is not possible under any circumstances to add expressions of different degree since they are not considered equivalent. Therefore, to solve the example that we have given previously, it will be necessary to add and operate with those of a similar degree. Those who do not have an even degree stay the same. For example, this would be the case of x4 since it is the only one that is raised to the fourth. The same would also happen with x3. Therefore, the first operation would be to add the negative expressions to the square, resulting in -4×2. We would do the same with the monomials and also with the first degree equations.
Therefore, the result that we would obtain from the example would be x4 + x3 -4×2 + 6 x -1. It is like a sum of natural numbers but taking into account the exponents that each of the equations has. Also remember that in the case of addition the commutative property can be applied. It does not matter if the first expression is in the first part than in the second, since the order of the factors does not alter the product. Yes, it would in the subtraction of polynomials since the second section receives a negative in all its signs, changing the positives to negatives and the negatives to positives. For this reason, if you put them in reverse order, there is a modification of the result that can throw an exam into the air.
Find the method that is most comfortable for you to add polynomials
Horizontal addition is not the only common way to add polynomials. There is also what is called vertical addition. In this case, the procedure is quite similar to what is usually taught in schools in order to start adding. From left to right, and one below the other, the expressions from higher to lower degree are placed, placing the expressions that are equivalent below. In the event that there was no homonymous degree, a zero would be put, since any number added or subtracted by zero remains absolutely the same. It only varies in multiplication where the product of a number times zero will always be zero. On the other hand, in a division it would not be possible to carry out the operation.
From this methodology, the sums of the equivalent expressions are made as we have done previously. For those who are beginning to work with polynomials, this second system is much more visual and easier to apply than the first. Therefore, when it comes to gaining agility when operating with polynomials, it is always more advisable to start with this method to gain confidence.
Don’t forget that the calculator allows you to add algebraic expressions and, therefore, when we add polynomials we can also use it to express the result directly. However, it is always advisable to start doing a new mathematical practice without using a calculator to understand what we are doing. With the passage of time, the elements are lost and it is more difficult to understand the meaning of each of the sections. Therefore, learning to add polynomials is strictly necessary for any math fan.