I. Expressions that are made up of numbers, variables, and their powers are called monomials using the multiplication action.
Examples of monomials:
and) a; b) ab; in) 12; d) -3c; e) 2a 2 ∙ (-3.5b) 3; e) -123.45xy 5 z; g) 8ac ∙ 2.5a 2 ∙ (-3c 3).
II. This type of monomial, when in the first place there is a numerical factor (coefficient), followed by variables with their degrees, is called the standard form of a monomial.
So, the monomials given above, under the letters a B C), d)and e)are written in standard form, and monomials under letters e) and g) it is required to bring to the standard form, that is, to such a form when the numerical factor is in the first place, and the letter factors with their indices are written down behind it, moreover, the letter factors are in alphabetical order. Here are the monomials e) and g) to the standard view.
e) 2a 2 ∙ (-3.5b) 3\u003d 2a 2 ∙ (-3.5) 3 ∙ b 3 \u003d -2a 2 ∙ 3.5 ∙ 3.5 ∙ 3.5 ∙ b 3 \u003d -85.75a 2 b 3;
g) 8ac ∙ 2.5a 2 ∙ (-3c 3)\u003d -8 ∙ 2.5 ∙ 3a 3 c 3 \u003d -60a 3 s 3.
III. The sum of the exponents of the degrees of all the variables included in the monomial is called the degree of the monomial.
Examples. What degree do monomials have a) - g)?
a) a. First;
b) ab. Second: and in the first degree and b in the first degree - the sum of indicators 1+1=2 ;
in) 12. Zero, since there are no alphabetic factors;
d) -3c.First;
e) -85.75a 2 b 3.Fifth. We have reduced this monomial to the standard form, we have and in the second degree and b in the third. Add up the indicators: 2+3=5 ;
e) -123.45xy 5 z. Seventh. We added the exponents of the letter multipliers: 1+5+1=7 ;
g) -60a 3 s 3.Sixth, since the sum of the alphabetic multipliers 3+3=6 .
IV. Monomials that have the same letter part are called similar monomials.
Example.Indicate similar monomials among the given monomials 1) -7).
1) 3aabbc; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 2 bac; 5) 10aaa 2 x; 6) -2.3a 4 x; 7) 34x 2 y.
Here are the monomials 1), 4) and 5) to the standard view. Then the line of these monomials will look like this:
1) 3a 2 b 2 c; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 3 bc; 5) 10a 4 x; 6) -2.3a 4 x; 7) 34x 2 y.
Similar will be those that have the same letter part, i.e. 1) and 3); 2) and 4); 5) and 6).
1) 3a 2 b 2 c and 3) 56a 2 b 2 c;
2) -4.1a 3 bc and 4) 98.7a 3 bc;
5) 10a 4 x and 6) -2.3a 4 x.
Monomial is an expression representing the product of two or more factors, each of which is a number expressed by a letter, numbers or a power (with a non-negative integer):
2a, a 3 x, 4abc, -7x
Since the product of the same factors can be written in the form of a degree, then a separately taken degree (with a non-negative integer exponent) is also a monomial:
(-4) 3 , x 5 ,
Since a number (integer or fractional), expressed in letter or numbers, can be written as the product of this number by one, then any separately taken number can also be considered as a monomial:
x, 16, -a,
Standard type of monomial
Standard type of monomial - is a monomial with only one numerical factor, which must be written in the first place. All variables are in alphabetical order and are contained in the monomial only once.
Numbers, variables and degrees of variables also refer to monomials of the standard form:
7, b, x 3 , -5b 3 z 2 - monomials of the standard type.
The numerical factor of a monomial of the standard form is called monomial coefficient... Monomial coefficients equal to 1 and -1 are usually not written.
If there is no numerical factor in the monomial of the standard form, then it is assumed that the coefficient of the monomial is 1:
x 3 \u003d 1 x 3
If there is no numerical factor in a monomial of the standard form and there is a minus sign in front of it, then it is assumed that the coefficient of the monomial is -1:
-x 3 \u003d -1 x 3
Reduction of a monomial to the standard form
To bring a monomial to a standard form, you need:
- Multiply numerical factors, if there are several. Raise a numeric factor to a power if it has an exponent. Put a numerical factor first.
- Multiply all the same variables so that each variable appears in the monomial only once.
- Arrange variables after a numeric factor in alphabetical order.
Example. Present a monomial in its standard form:
a) 3 yx 2 (-2) y 5 x; b) 6 bc 0.5 ab 3
Decision:
a) 3 yx 2 (-2) y 5 x \u003d 3 (-2) x 2 xyy 5 = -6x 3 y 6
b) 6 bc 0.5 ab 3 \u003d 6 0.5 abb 3 c = 3ab 4 c
Monomial degree
Monomial degree is the sum of the exponents of all the letters included in it.
If a monomial is a number, that is, it does not contain variables, then its degree is considered equal to zero. For instance:
5, -7, 21 - zero degree monomials.
Therefore, in order to find the degree of a monomial, you need to determine the exponent of each of the letters included in it and add these indicators. If the letter exponent is not specified, then it is equal to one.
Examples:
So how are u x the exponent is not specified, so it is equal to 1. The monomial does not contain other variables, so its degree is 1.
The monomial contains only one variable in the second degree, so the degree of this monomial is 2.
3) ab 3 c 2 d
Index a equals 1, exponent b - 3, indicator c - 2, indicator d - 1. The degree of a given monomial is equal to the sum of these indicators.
Monomials are products of numbers, variables and their powers. Numbers, variables and their degrees are also considered monomials. For example: 12ac, -33, a ^ 2b, a, c ^ 9. The monomial 5aa2b2b can be reduced to the form 20a ^ 2b ^ 2. This form is called the standard form of a monomial. That is, the standard form of a monomial is the product of the coefficient (in the first place) and the degrees of the variables. Coefficients 1 and -1 are not written, but they keep minus from -1. Monomial and its standard form
Expressions 5a2x, 2a3 (-3) x2, b2x are products of numbers, variables and their powers. Such expressions are called monomials. Numbers, variables and their degrees are also considered monomials.
For example, expressions - 8, 35, y and y2 - are monomials.
The standard form of a monomial is a monomial in the form of the product of a numerical factor in the first place and the degrees of various variables. Any monomial can be reduced to a standard form by multiplying all variables and numbers included in it. Here is an example of reducing a monomial to a standard form:
4x2y4 (-5) yx3 \u003d 4 (-5) x2x3y4y \u003d -20x5y5
The numerical factor of a monomial written in the standard form is called the coefficient of a monomial. For example, the coefficient of a monomial -7x2y2 is -7. The coefficients of the monomials x3 and -xy are considered equal to 1 and -1, since x3 \u003d 1x3 and -xy \u003d -1xy
The degree of a monomial is the sum of the exponents of all the variables included in it. If a monomial does not contain variables, that is, is a number, then its degree is considered equal to zero.
For example, the degree of the monomial 8x3yz2 is 6, the monomial 6x is 1, the monomial -10 is 0.
Multiplication of monomials. Exponentiation of monomials
When multiplying monomials and raising monomials to a power, the rule for multiplying powers with the same base and the rule for raising a power to a power are used. In this case, a monomial is obtained, which is usually represented in a standard form.
for instance
4x3y2 (-3) x2y \u003d 4 (-3) x3x2y2y \u003d -12x5y3
((-5) x3y2) 3 \u003d (-5) 3x3 * 3y2 * 3 \u003d -125x9y6
We noted that any monomial can be bring to standard form... In this article, we will figure out what is called reducing a monomial to a standard form, what actions allow this process to be carried out, and consider the solutions of examples with detailed explanations.
Page navigation.
What does it mean to bring a monomial to a standard form?
It is convenient to work with monomials when they are written in a standard form. However, monomials are often given in a form other than the standard one. In these cases, one can always go from the original monomial to a monomial of the standard form by performing identical transformations. The process of carrying out such transformations is called the reduction of a monomial to a standard form.
Let us generalize the above reasoning. Bring the monomial to the standard form - it means to perform such identical transformations with it so that it takes a standard form.
How to bring a monomial to a standard form?
It's time to figure out how to bring monomials to the standard form.
As is known from the definition, non-standard monomials are products of numbers, variables and their degrees, and, possibly, repeating. A monomial of the standard form can contain only one number and non-repeating variables or their degrees in its record. Now it remains to understand how to bring works of the first type to the form of the second?
To do this, you need to use the following the rule for reducing the monomial to the standard formconsisting of two steps:
- First, the grouping of numerical factors, as well as the same variables and their degrees is performed;
- Second, the product of the numbers is calculated and applied.
As a result of applying the voiced rule, any monomial will be reduced to a standard form.
Examples, solutions
It remains to learn how to apply the rule from the previous paragraph when solving examples.
Example.
Reduce the monomial 3 · x · 2 · x 2 to its standard form.
Decision.
Let's group the numerical factors and factors with the variable x. After the grouping, the original monomial takes the form (3 2) (x x 2). The product of the numbers in the first parentheses is 6, and the rule for multiplying powers with the same base allows the expression in the second parentheses to be represented as x 1 + 2 \u003d x 3. As a result, we get a polynomial of the standard form 6 · x 3.
Here is a short record of the solution: 3 x 2 x 2 \u003d (3 2) (x x 2) \u003d 6 x 3.
Answer:
3 x 2 x 2 \u003d 6 x 3.
So, to bring a monomial to a standard form, you need to be able to group factors, multiply numbers, and work with powers.
To consolidate the material, we will solve one more example.
Example.
Present the monomial in standard form and indicate its coefficient.
Decision.
The original monomial has a unique numerical factor −1 in its notation, we move it to the beginning. After that, we separately group the factors with the variable a, separately - with the variable b, and there is nothing to group the variable m with, we leave it as it is, we have 
... After performing actions with degrees in brackets, the monomial will take the standard form we need, from which we can see the coefficient of the monomial equal to −1. Minus one can be replaced with a minus sign:.