What does a monomial mean in standard form. I

Monomials are one of the main types of expressions studied within school course algebra. In this article we will tell you what these expressions are, we will define them standard view and show examples, as well as deal with related concepts, such as the degree of a monomial and its coefficient.

What is a monomial

School textbooks usually give the following definition of this concept:

Definition 1

Monomers include numbers, variables, as well as their degrees with natural indicator and different types works made from them.

Based on this definition, we can give examples of such expressions. So, all numbers 2 , 8 , 3004 , 0 , - 4 , - 6 , 0 , 78 , 1 4 , - 4 3 7 will refer to monomials. All variables, for example, x , a , b , p , q , t , y , z will also be monomials by definition. This also includes the powers of variables and numbers, for example, 6 3 , (− 7 , 41) 7 , x 2 and t 15, as well as expressions like 65 x , 9 (− 7) x y 3 6 , x x y 3 x y 2 z etc. Please note that a monomial can include either one number or variable, or several, and they can be mentioned several times as part of one polynomial.

Such types of numbers as integers, rationals, naturals also belong to monomials. It can also include real and complex numbers. So, expressions like 2 + 3 i x z 4 , 2 x , 2 π x 3 will also be monomials.

What is the standard form of a monomial and how to convert an expression to it

For convenience of work, all monomials are first reduced to a special form, called the standard one. Let's be specific about what this means.

Definition 2

The standard form of the monomial call it such a form in which it is the product of a numerical factor and natural degrees different variables. The numerical factor, also called the monomial coefficient, is usually written first from the left side.

For clarity, we select several monomials of the standard form: 6 (this is a monomial without variables), 4 · a , − 9 · x 2 · y 3 , 2 3 5 · x 7 . This also includes the expression x y(here the coefficient will be equal to 1), − x 3(here the coefficient is - 1).

Now we give examples of monomials that need to be brought to standard form: 4 a a 2 a 3(here you need to combine the same variables), 5 x (− 1) 3 y 2(here you need to combine the numerical factors on the left).

Usually, in the case when a monomial has several variables written in letters, the letter factors are written in alphabetical order. For example, the preferred entry 6 a b 4 c z 2, how b 4 6 a z 2 c. However, the order may be different if the purpose of the computation requires it.

Any monomial can be reduced to standard form. To do this, you need to perform all the necessary identical transformations.

The concept of the degree of a monomial

Very important is related concept monomial degree. Let us write down the definition of this concept.

Definition 3

Degree of a monomial, written in standard form, is the sum of the exponents of all variables that are included in its record. If there is not a single variable in it, and the monomial itself is different from 0, then its degree will be zero.

Let us give examples of the degrees of the monomial.

Example 1

So, monomial a has degree 1 because a = a 1 . If we have a monomial 7 , then it will have a zero degree, since it has no variables and is different from 0 . And here is the entry 7 a 2 x y 3 a 2 will be a monomial of the 8th degree, because the sum of the exponents of all the degrees of the variables included in it will be equal to 8: 2 + 1 + 3 + 2 = 8 .

The standardized monomial and the original polynomial will have the same degree.

Example 2

Let's show how to calculate the degree of a monomial 3 x 2 y 3 x (− 2) x 5 y. In standard form, it can be written as − 6 x 8 y 4. We calculate the degree: 8 + 4 = 12 . Hence, the degree of the original polynomial is also equal to 12 .

The concept of a monomial coefficient

If we have a standardized monomial that includes at least one variable, then we talk about it as a product with one numerical factor. This factor is called the numerical coefficient, or the monomial coefficient. Let's write down the definition.

Definition 4

The coefficient of a monomial is the numerical factor of a monomial reduced to standard form.

Take, for example, the coefficients of various monomials.

Example 3

So, in the expression 8 a 3 the coefficient will be the number 8, and in (− 2 , 3) ​​x y z they will − 2 , 3 .

Particular attention should be paid to the ratios equal to one and minus one. As a rule, they are not explicitly indicated. It is believed that in a monomial of the standard form, in which there is no numerical factor, the coefficient is 1, for example, in the expressions a, x z 3, a t x, since they can be considered as 1 a, x z 3 - as 1 x z 3 etc.

Similarly, in monomials that do not have a numerical factor and that begin with a minus sign, we can consider the coefficient - 1.

Example 4

For example, the expressions − x, − x 3 y z 3 will have such a coefficient, since they can be represented as − x = (− 1) x, − x 3 y z 3 = (− 1) x 3 y z 3 etc.

If a monomial does not have a single literal multiplier at all, then it is possible to talk about a coefficient in this case as well. The coefficients of such monomials-numbers will be these numbers themselves. So, for example, the coefficient of the monomial 9 will be equal to 9.

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Lesson on the topic: "Standard form of a monomial. Definition. Examples"

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Monomial. Definition

Monomial is a mathematical expression that represents the product prime factor and one or more variables.

Monomials include all numbers, variables, their powers with a natural exponent:
42; 3; 0; 62; 2 3 ; b 3 ; ax4; 4x3; 5a2; 12xyz 3 .

Quite often it is difficult to determine whether a given mathematical expression refers to a monomial or not. For example, $\frac(4a^3)(5)$. Is it monomial or not? To answer this question, we need to simplify the expression, i.e. represent in the form: $\frac(4)(5)*а^3$.
We can say for sure that this expression is a monomial.

Standard form of a monomial

When calculating, it is desirable to bring the monomial to the standard form. This is the shortest and most understandable notation of the monomial.

The order of bringing the monomial to the standard form is as follows:
1. Multiply the coefficients of the monomial (or numerical factors) and put the result in the first place.
2. Select all degrees with the same letter base and multiply them.
3. Repeat point 2 for all variables.

Examples.
I. Reduce the given monomial $3x^2zy^3*5y^2z^4$ to standard form.

Decision.
1. Multiply the coefficients of the monomial $15x^2y^3z * y^2z^4$.
2. Now we present like terms$15x^2y^5z^5$.

II. Convert the given monomial $5a^2b^3 * \frac(2)(7)a^3b^2c$ to standard form.

Decision.
1. Multiply the coefficients of the monomial $\frac(10)(7)a^2b^3*a^3b^2c$.
2. Now let us present similar terms $\frac(10)(7)a^5b^5c$.