After studying monomials, we move on to polynomials. This article will tell you about everyone necessary information, necessary to perform actions on them. We will define a polynomial with accompanying definitions term of a polynomial, that is, free and similar, consider a polynomial of a standard form, introduce a degree and learn how to find it, work with its coefficients.
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Polynomial and its terms - definitions and examples
The definition of a polynomial was necessary back in 7 class after studying monomials. Let's look at its full definition.
Definition 1
Polynomial the sum of monomials is considered, and the monomial itself is special case polynomial.
From the definition it follows that examples of polynomials can be different: 5 , 0 , − 1 , x, 5 a b 3, x 2 · 0 , 6 · x · (− 2) · y 12 , - 2 13 · x · y 2 · 3 2 3 · x · x 3 · y · z and so on. From the definition we have that 1+x, a 2 + b 2 and the expression x 2 - 2 x y + 2 5 x 2 + y 2 + 5, 2 y x are polynomials.
Let's look at some more definitions.
Definition 2
Members of the polynomial its constituent monomials are called.
Consider an example where we have a polynomial 3 x 4 − 2 x y + 3 − y 3, consisting of 4 terms: 3 x 4, − 2 x y, 3 and − y 3. Such a monomial can be considered a polynomial, which consists of one term.
Definition 3
Polynomials that contain 2, 3 trinomials have the corresponding name - binomial And trinomial.
It follows that an expression of the form x+y– is a binomial, and the expression 2 x 3 q − q x x x + 7 b is a trinomial.
By school curriculum worked with a linear binomial of the form a · x + b, where a and b are some numbers, and x is a variable. Let's consider examples of linear binomials of the form: x + 1, x 7, 2 − 4 with examples square trinomials x 2 + 3 x − 5 and 2 5 x 2 - 3 x + 11 .
To transform and solve it is necessary to find and bring similar terms. For example, a polynomial of the form 1 + 5 x − 3 + y + 2 x has similar terms 1 and - 3, 5 x and 2 x. They are divided into special group called similar terms of a polynomial.
Definition 4
Similar terms of a polynomial are similar terms found in a polynomial.
In the example above, we have that 1 and - 3, 5 x and 2 x are similar terms of the polynomial or similar terms. In order to simplify the expression, find and reduce similar terms.
Polynomial of standard form
All monomials and polynomials have their own specific names.
Definition 5
Polynomial of standard form is a polynomial in which each term included in it has a monomial of standard form and does not contain similar terms.
From the definition it is clear that it is possible to reduce polynomials of the standard form, for example, 3 x 2 − x y + 1 and __formula__, and the entry is in standard form. The expressions 5 + 3 · x 2 − x 2 + 2 · x · z and 5 + 3 · x 2 − x 2 + 2 · x · z are not polynomials of standard form, since the first of them has similar terms in the form 3 · x 2 and − x 2, and the second contains a monomial of the form x · y 3 · x · z 2, which differs from the standard polynomial.
If circumstances require it, sometimes the polynomial is reduced to a standard form. The concept of a free term of a polynomial is also considered a polynomial of standard form.
Definition 6
Free term of a polynomial is a polynomial of standard form that does not have a literal part.
In other words, when a polynomial in standard form has a number, it is called a free member. Then the number 5 is the free term of the polynomial x 2 z + 5, and the polynomial 7 a + 4 a b + b 3 does not have a free term.
Degree of a polynomial - how to find it?
The definition of the degree of a polynomial itself is based on the definition of a standard form polynomial and on the degrees of the monomials that are its components.
Definition 7
Degree of a polynomial of standard form is called the largest of the degrees included in its notation.
Let's look at an example. The degree of the polynomial 5 x 3 − 4 is equal to 3, because the monomials included in its composition have degrees 3 and 0, and the larger of them is 3, respectively. The definition of the degree from the polynomial 4 x 2 y 3 − 5 x 4 y + 6 x is equal to the largest of the numbers, that is, 2 + 3 = 5, 4 + 1 = 5 and 1, which means 5.
It is necessary to find out how the degree itself is found.
Definition 8
Polynomial degree any number is the degree of the corresponding polynomial in standard form.
When a polynomial is not written in standard form, but you need to find its degree, you need to reduce it to the standard form, and then find the required degree.
Example 1
Find the degree of a polynomial 3 a 12 − 2 a b c c a c b + y 2 z 2 − 2 a 12 − a 12.
Solution
First, let's present the polynomial in standard form. We get an expression of the form:
3 a 12 − 2 a b c c a c b + y 2 z 2 − 2 a 12 − a 12 = = (3 a 12 − 2 a 12 − a 12) − 2 · (a · a) · (b · b) · (c · c) + y 2 · z 2 = = − 2 · a 2 · b 2 · c 2 + y 2 · z 2
When obtaining a polynomial of standard form, we find that two of them stand out clearly - 2 · a 2 · b 2 · c 2 and y 2 · z 2 . To find the degrees, we count and find that 2 + 2 + 2 = 6 and 2 + 2 = 4. It can be seen that the largest of them is 6. From the definition it follows that 6 is the degree of the polynomial − 2 · a 2 · b 2 · c 2 + y 2 · z 2 , and therefore the original value.
Answer: 6 .
Coefficients of polynomial terms
Definition 9When all terms of a polynomial are monomials of the standard form, then in this case they have the name coefficients of polynomial terms. In other words, they can be called coefficients of the polynomial.
When considering the example, it is clear that a polynomial of the form 2 x − 0, 5 x y + 3 x + 7 contains 4 polynomials: 2 x, − 0, 5 x y, 3 x and 7 with their corresponding coefficients 2, − 0, 5, 3 and 7. This means that 2, − 0, 5, 3 and 7 are considered coefficients of terms of a given polynomial of the form 2 x − 0, 5 x y + 3 x + 7. When converting, it is important to pay attention to the coefficients in front of the variables.
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On this lesson we will recall the basic definitions of this topic and consider some typical problems, namely, reducing a polynomial to a standard form and calculating the numerical value of given values variables. We will solve several examples in which reduction to standard form will be used to solve various kinds tasks.
Subject:Polynomials. Arithmetic operations on monomials
Lesson:Reducing a polynomial to standard form. Typical tasks
Let us recall the basic definition: a polynomial is the sum of monomials. Each monomial that is part of a polynomial as a term is called its member. For example:
Binomial;
Polynomial;
Binomial;
Since a polynomial consists of monomials, the first action with a polynomial follows from here - you need to bring all monomials to a standard form. Let us remind you that for this you need to multiply all the numerical factors - get numerical coefficient, and multiply the corresponding degrees - get the letter part. In addition, let us pay attention to the theorem about the product of powers: when powers are multiplied, their exponents add up.
Let's consider an important operation - reducing a polynomial to standard form. Example:
Comment: to bring a polynomial to a standard form, you need to bring all the monomials included in its composition to a standard form, after which, if there are similar monomials - and these are monomials with the same letter part - perform actions with them.
So, we looked at the first typical problem - bringing a polynomial to a standard form.
Next typical task- calculation specific meaning polynomial for given numerical values the variables included in it. Let's continue to look at the previous example and set the values of the variables:
Comment: recall that a unit in any natural degree equal to one, and zero to any natural power equal to zero, in addition, recall that when multiplying any number by zero, we get zero.
Let's look at a number of examples of typical operations of bringing a polynomial to a standard form and calculating its value:
Example 1 - bring to standard form:
Comment: the first step is to bring the monomials to standard form, you need to bring the first, second and sixth; second action - we present similar members, that is, we perform the given tasks on them arithmetic operations: we add the first with the fifth, the second with the third, the rest are rewritten without changes, since they have no similar ones.
Example 2 - calculate the value of the polynomial from example 1 given the values of the variables:
Comment: when calculating, you should remember that a unit to any natural power is one; if it is difficult to calculate powers of two, you can use the table of powers.
Example 3 - instead of an asterisk, put a monomial such that the result does not contain a variable:
Comment: regardless of the task, the first action is always the same - bring the polynomial to a standard form. In our example, this action comes down to bringing similar terms. After this, you should carefully read the condition again and think about how we can get rid of the monomial. Obviously, for this you need to add the same monomial to it, but with opposite sign- . Next, we replace the asterisk with this monomial and make sure that our solution is correct.
In this lesson, we will recall the basic definitions of this topic and consider some typical problems, namely, reducing a polynomial to a standard form and calculating a numerical value for given values of variables. We will solve several examples in which reduction to a standard form will be used to solve various types of problems.
Subject:Polynomials. Arithmetic operations on monomials
Lesson:Reducing a polynomial to standard form. Typical tasks
Let us recall the basic definition: a polynomial is the sum of monomials. Each monomial that is part of a polynomial as a term is called its member. For example:
Binomial;
Polynomial;
Binomial;
Since a polynomial consists of monomials, the first action with a polynomial follows from here - you need to bring all monomials to a standard form. Let us remind you that to do this you need to multiply all the numerical factors - get a numerical coefficient, and multiply the corresponding powers - get the letter part. In addition, let us pay attention to the theorem about the product of powers: when powers are multiplied, their exponents add up.
Let's consider an important operation - reducing a polynomial to standard form. Example:
Comment: to bring a polynomial to a standard form, you need to bring all the monomials included in its composition to a standard form, after which, if there are similar monomials - and these are monomials with the same letter part - perform actions with them.
So, we looked at the first typical problem - bringing a polynomial to a standard form.
The next typical problem is calculating the specific value of a polynomial for given numerical values of the variables included in it. Let's continue to look at the previous example and set the values of the variables:
Comment: let us recall that one to any natural power is equal to one, and zero to any natural power is equal to zero, in addition, we recall that when multiplying any number by zero, we get zero.
Let's look at a number of examples of typical operations of bringing a polynomial to a standard form and calculating its value:
Example 1 - bring to standard form:
Comment: the first step is to bring the monomials to standard form, you need to bring the first, second and sixth; second action - we bring similar terms, that is, we perform the given arithmetic operations on them: we add the first with the fifth, the second with the third, we rewrite the rest without changes, since they have no similar ones.
Example 2 - calculate the value of the polynomial from example 1 given the values of the variables:
Comment: when calculating, you should remember that a unit to any natural power is one; if it is difficult to calculate powers of two, you can use the table of powers.
Example 3 - instead of an asterisk, put a monomial such that the result does not contain a variable:
Comment: regardless of the task, the first action is always the same - bring the polynomial to a standard form. In our example, this action comes down to bringing similar terms. After this, you should carefully read the condition again and think about how we can get rid of the monomial. Obviously, to do this, you need to add the same monomial to it, but with the opposite sign - . Next, we replace the asterisk with this monomial and make sure that our solution is correct.
We said that there are both standard and non-standard polynomials. There we noted that anyone can bring the polynomial to standard form. In this article, we will first find out what meaning this phrase carries. Next we list the steps that allow you to transform any polynomial into standard view. Finally, let's look at solutions typical examples. We will describe the solutions in great detail in order to understand all the nuances that arise when reducing polynomials to standard form.
Page navigation.
What does it mean to reduce a polynomial to standard form?
First you need to clearly understand what is meant by reducing a polynomial to standard form. Let's figure this out.
Polynomials, like any other expressions, can be subjected to identical transformations. As a result of performing such transformations, expressions are obtained that are identically equal to the original expression. Thus, performing certain transformations with polynomials of non-standard form allows one to move on to polynomials that are identically equal to them, but written in standard form. This transition is called reducing the polynomial to standard form.
So, reduce the polynomial to standard form- this means replacing the original polynomial with an identically equal polynomial of a standard form, obtained from the original one by carrying out identical transformations.
How to reduce a polynomial to standard form?
Let's think about what transformations will help us bring the polynomial to a standard form. We will start from the definition of a standard form polynomial.
By definition, every term of a polynomial of standard form is a monomial of standard form, and a polynomial of standard form contains no similar terms. In turn, polynomials written in a form other than the standard one can consist of monomials in a non-standard form and can contain similar terms. This logically follows next rule, explaining how to reduce a polynomial to standard form:
- first you need to bring the monomials that make up the original polynomial to standard form,
- then perform the reduction of similar terms.
As a result, a polynomial of standard form will be obtained, since all its terms will be written in standard form, and it will not contain similar terms.
Examples, solutions
Let's look at examples of reducing polynomials to standard form. When solving, we will follow the steps dictated by the rule from the previous paragraph.
Here we note that sometimes all the terms of a polynomial are immediately written in standard form; in this case, it is enough to just give similar terms. Sometimes, after reducing the terms of a polynomial to a standard form, there are no similar terms, therefore, the stage of bringing similar terms is omitted in this case. IN general case you have to do both.
Example.
Present the polynomials in standard form: 5 x 2 y+2 y 3 −x y+1 , 0.8+2 a 3 0.6−b a b 4 b 5 And .
Solution.
All terms of the polynomial 5·x 2 ·y+2·y 3 −x·y+1 are written in standard form; it does not have similar terms, therefore, this polynomial is already presented in standard form.
Let's move on to the next polynomial 0.8+2 a 3 0.6−b a b 4 b 5. Its form is not standard, as evidenced by the terms 2·a 3 ·0.6 and −b·a·b 4 ·b 5 of a non-standard form. Let's present it in standard form.
At the first stage of bringing the original polynomial to standard form, we need to present all its terms in standard form. Therefore, we reduce the monomial 2·a 3 ·0.6 to standard form, we have 2·a 3 ·0.6=1.2·a 3 , after which we take the monomial −b·a·b 4 ·b 5 , we have −b·a·b 4 ·b 5 =−a·b 1+4+5 =−a·b 10. Thus, . In the resulting polynomial, all terms are written in standard form; moreover, it is obvious that there are no similar terms in it. Consequently, this completes the reduction of the original polynomial to standard form.
It remains to present the last of the given polynomials in standard form. After bringing all its members to standard form, it will be written as 
. It has similar members, so you need to cast similar members: 
So the original polynomial took the standard form −x·y+1.
Answer:
5 x 2 y+2 y 3 −x y+1 – already in standard form, 0.8+2 a 3 0.6−b a b 4 b 5 =0.8+1.2 a 3 −a b 10, .
Often, bringing a polynomial to a standard form is only an intermediate step in answering the question posed to the problem. For example, finding the degree of a polynomial requires its preliminary representation in standard form.
Example.
Give a polynomial 
to the standard form, indicate its degree and arrange the terms in descending degrees of the variable.
Solution.
First, we bring all the terms of the polynomial to standard form: 
.
Now we present similar terms: 
So we brought the original polynomial to standard form, this allows us to determine the degree of the polynomial, which is equal to the highest to a greater extent monomials included in it. Obviously it is equal to 5.
It remains to arrange the terms of the polynomial in decreasing powers of the variables. To do this, you just need to rearrange the terms in the resulting polynomial of standard form, taking into account the requirement. Greatest degree has a term z 5, the degrees of the terms , −0.5·z 2 and 11 are equal to 3, 2 and 0, respectively. Therefore, a polynomial with terms arranged in decreasing powers of the variable will have the form 
.
Answer:
The degree of the polynomial is 5, and after arranging its terms in descending degrees of the variable, it takes the form .
Bibliography.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Mordkovich A. G. Algebra. 7th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
- Algebra and started mathematical analysis. 10th grade: textbook. for general education institutions: basic and profile. levels / [Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; edited by A. B. Zhizhchenko. - 3rd ed. - M.: Education, 2010.- 368 p. : ill. - ISBN 978-5-09-022771-1.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
For example, expressions:
a - b + c, x 2 - y 2 , 5x - 3y - z- polynomials
The monomials that make up a polynomial are called members of the polynomial. Consider the polynomial:
7a + 2b - 3c - 11
expressions: 7 a, 2b, -3c and -11 are the terms of the polynomial. Note that the -11 member does not contain a variable; such members consisting only of a number are called free.
It is generally accepted that any monomial is a special case of a polynomial, consisting of one term. In this case, a monomial is the name for a polynomial with one term. For polynomials consisting of two and three members, there is also special names- binomial and trinomial, respectively:
7a- monomial
7a + 2b- binomial
7a + 2b - 3c- trinomial
Similar members
Similar members- monomials included in a polynomial that differ from each other only by coefficient, sign, or do not differ at all (opposite monomials can also be called similar). For example, in a polynomial:
| 3a 2 b | + | 5abc 2 | + | 2a 2 b | - | 7abc 2 | - | 2a 2 b |
members 3 a 2 b, 2a 2 b and 2 a 2 b, as well as members 5 abc 2 and -7 abc 2 are similar terms.
Bringing similar members
If a polynomial contains similar terms, then it can be reduced to more simple view by combining similar members into one. This action is called bringing similar members. First of all, let’s enclose all similar terms separately in brackets:
(3a 2 b + 2a 2 b - 2a 2 b) + (5abc 2 - 7abc 2)
To combine several similar monomials into one, you need to add their coefficients and leave the letter factors unchanged:
((3 + 2 - 2)a 2 b) + ((5 - 7)abc 2) = (3a 2 b) + (-2abc 2) = 3a 2 b - 2abc 2
Coercion of similar terms is a replacement operation algebraic sum several similar monomials by one monomial.
Polynomial of standard form
Polynomial of standard form is a polynomial all of whose terms are monomials of standard form, among which there are no similar terms.
To bring a polynomial to standard form, it is enough to reduce similar terms. For example, represent the expression as a polynomial of the standard form:
3xy + x 3 - 2xy - y + 2x 3
First, let's find similar terms:
If all the terms of a standard type polynomial contain the same variable, then its terms are usually arranged from greatest to least degree. Free member polynomial, if there is one, is placed on last place- on right.
For example, a polynomial
3x + x 3 - 2x 2 - 7
should be written like this:
x 3 - 2x 2 + 3x - 7
