Here I will explain the basics you need to know about polynomials. You will learn what a monomial is, what a polynomial is and how to calculate its degree and numerical value.
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What is a monomial
A monomial is an algebraic expression consisting of a sign (positive or negative), numbers and variables, which are multiplied together, and therefore form a product.
In turn, the variables may be elevated to a natural exponent.
Let’s take a look at some examples of monomials to help you understand it better:
Next, let’s take a closer look at the elements of a monomial, since each of them is named in a certain way.
Elements of a monomial
When naming the elements of a monomial, they are divided into two parts: the coefficient and the literal part.
The coefficient is formed by the sign and the numbers and the literal part by the variables and their exponents.
Let’s take a closer look at each of them:
As I have just indicated, the coefficient is formed by the sign and the number, which is multiplying the literal part. It is usually placed at the beginning. To see it more clearly, I mark the coefficients of the previous examples in red:
If the coefficient has no sign, it is equivalent to a positive sign, as is the case with:
What if no coefficient appears?
If no coefficient is displayed, such as:
It means that in this case the coefficient is 1 and is not written because you know that multiplying a value by 1 does not change its result:
On the other hand, the coefficient can never be zero, since the whole expression would have a value of 0:
To finish with the coefficient, indicate that it does not have to be an integer, it can also be a rational or irrational number:
The literal part is formed by the variables with their exponents. Maro now in blue the literal part of the previous examples:
If any variable has no exponent, it is equivalent to that variable being raised to 1, as is the case with:
Grade of a monomial
Now that we are more familiar with the monomial we will see how to calculate its degree.
The degree of a monomial is the sum of all the exponents of its variables, regardless of whether the variables are the same or not.
Calculating the degree of a monomial has nothing to do with the multiplication of powers of the same base, in which the exponents are added together and it is a necessary condition that they have the same base in order for the exponents to be added together.
This is different, don’t get confused. Here the exponents of all its variables are added together.
Let’s calculate the degree of the previous monomials:
What is a polynomial
A polynomial is the algebraic sum of two or more monomials. Each monomial of the polynomial is called a term. This is an example of a polynomial:
Each term of the polynomial, as we have seen before, is composed of coefficient and literal part.
The polynomial of a single term is called monomial, the one of two binomials, the one of three trinomials…
On the other hand, when polynomials only have one variable they are usually indicated by naming them with the letter P onwards (although this does not matter) and enclosing the variable they have in brackets:
Grade of a polynomial
The degree of a polynomial is the degree of its highest monomial. Let’s look at it with an example:
In this case, the first monomial has degree 6, the second degree 11 and the third degree 5, therefore, the degree of the polynomial is 11, since it is the highest of the three degrees.
If the polynomial has only one variable, its degree is calculated exactly the same:
The degree of the polynomial coincides with the term of highest degree.
Numerical value of a polynomial
Numerical value of a polynomial
The numerical value of a polynomial is the value it takes when variables are replaced by a given value.
Once the values that the variables take are known, we only need to replace the variables with the numbers in the polynomial and then operate to find their value.
Let’s look at it with an example: Find the numerical value of the next polynomial:
For these variable values:
To calculate the value of the polynomial, replace the x with -1 and the y with 2 and operate:
The value of the polynomial in this case will be -50.
To calculate the numerical value of polynomials that depend on a single variable, we do the same thing. For example, calculate the value of the next polynomial:
We replace the x with -2 and operate:
The value of the polynomial in this case is -244.
Be very careful with the minus signs in the powers.