We learn to reduce polynomials to standard form.

The concept of a polynomial

Definition of polynomial: A polynomial is the sum of monomials. Polynomial example:

here we see the sum of two monomials, and this is a polynomial, i.e. sum of monomials.

The terms that make up a polynomial are called terms of the polynomial.

Is the difference of monomials a polynomial? Yes, it is, because the difference is easily reduced to a sum, example: 5a – 2b = 5a + (-2b).

Monomials are also considered polynomials. But a monomial has no sum, then why is it considered a polynomial? And you can add zero to it and get its sum with a zero monomial. So the monomial is special case polynomial, it consists of one member.

The number zero is the zero polynomial.

Standard form of polynomial

What is a polynomial of standard form? A polynomial is the sum of monomials, and if all these monomials that make up the polynomial are written in standard form, and there should be no similar ones among them, then the polynomial is written in standard form.

An example of a polynomial in standard form:

here the polynomial consists of 2 monomials, each of which has a standard form; among the monomials there are no similar ones.

Now an example of a polynomial that does not have a standard form:

here two monomials: 2a and 4a are similar. You need to add them up, then the polynomial will take the standard form:

Another example:

This polynomial is reduced to standard view? No, his second term is not written in standard form. Writing it in standard form, we obtain a polynomial of standard form:

Polynomial degree

What is the degree of a polynomial?

Polynomial degree definition:

The degree of a polynomial is the highest degree that the monomials that make up a given polynomial of standard form have.

Example. What is the degree of the polynomial 5h? The degree of the polynomial 5h is equal to one, because this polynomial contains only one monomial and its degree is equal to one.

Another example. What is the degree of the polynomial 5a 2 h 3 s 4 +1? The degree of the polynomial 5a 2 h 3 s 4 + 1 is equal to nine, because this polynomial includes two monomials, the first monomial 5a 2 h 3 s 4 has the highest degree, and its degree is 9.

Another example. What is the degree of the polynomial 5? The degree of a polynomial 5 is zero. So, the degree of a polynomial consisting only of a number, i.e. without letters, equals zero.

The last example. What is the degree of the zero polynomial, i.e. zero? The degree of the zero polynomial is not defined.

A polynomial is the sum of monomials. If all the terms of a polynomial are written in standard form (see paragraph 51) and similar terms are reduced, you will get a polynomial of standard form.

Any integer expression can be converted into a polynomial of standard form - this is the purpose of transformations (simplifications) of integer expressions.

Let's look at examples in which an entire expression needs to be reduced to the standard form of a polynomial.

Solution. First, let's bring the terms of the polynomial to standard form. We obtain After bringing similar terms, we obtain a polynomial of the standard form

Solution. If there is a plus sign in front of the brackets, then the brackets can be omitted, preserving the signs of all terms enclosed in brackets. Using this rule for opening parentheses, we get:

Solution. If the parentheses are preceded by a minus sign, then the parentheses can be omitted by changing the signs of all terms enclosed in the brackets. Using this rule for hiding parentheses, we get:

Solution. The product of a monomial and a polynomial, according to the distributive law, is equal to the sum of the products of this monomial and each member of the polynomial. We get

Solution. We have

It remains to give similar terms (they are underlined). We get:

53. Abbreviated multiplication formulas.

In some cases, bringing an entire expression to the standard form of a polynomial is carried out using the identities:

These identities are called abbreviated multiplication formulas,

Let's look at examples in which you need to convert a given expression into standard form myogochlea.

Example 1. .

Solution. Using formula (1), we obtain:

Example 2. .

Solution.

Example 3...

Solution. Using formula (3), we obtain:

Example 4.

Solution. Using formula (4), we obtain:

54. Factoring polynomials.

Sometimes you can transform a polynomial into a product of several factors - polynomials or subnomials. Such an identity transformation is called factorization of a polynomial. In this case, the polynomial is said to be divisible by each of these factors.

Let's look at some ways to factor polynomials,

1) Taking the common factor out of brackets. This transformation is a direct consequence of the distributive law (for clarity, you just need to rewrite this law “from right to left”):

Example 1: Factor a polynomial

Solution. .

Usually, when taking the common factor out of brackets, each variable included in all terms of the polynomial is taken out with the lowest exponent that it has in this polynomial. If all the coefficients of the polynomial are integers, then the largest in modulus is taken as the coefficient of the common factor common divisor all coefficients of the polynomial.

2) Using abbreviated multiplication formulas. Formulas (1) - (7) from paragraph 53, being read from right to left, in many cases turn out to be useful for factoring polynomials.

Example 2. Factorize.

Solution. We have. Applying formula (1) (difference of squares), we obtain . By applying

Now formulas (4) and (5) (sum of cubes, difference of cubes), we get:

Example 3...

Solution. First, let's take the common factor out of the bracket. To do this, we will find the greatest common divisor of the coefficients 4, 16, 16 and the smallest exponents with which the variables a and b are included in the constituent monomials of this polynomial. We get:

3) Method of grouping. It is based on the fact that the commutative and associative laws of addition allow the members of a polynomial to be grouped in various ways. Sometimes it is possible to group in such a way that after taking the common factors out of brackets, the same polynomial remains in brackets in each group, which in turn, as a common factor, can be taken out of brackets. Let's look at examples of factoring a polynomial.

Example 4. .

Solution. Let's group as follows:

In the first group, let's take the common factor out of the brackets into the second - the common factor 5. We get Now we put the polynomial as a common factor out of the brackets: Thus, we get:

Example 5.

Solution. .

Example 6.

Solution. Here, no grouping will lead to the appearance of the same polynomial in all groups. In such cases, it is sometimes useful to represent a member of the polynomial as a sum, and then try the grouping method again. In our example, it is advisable to represent it as a sum. We get

Example 7.

Solution. Add and subtract a monomial We get

55. Polynomials in one variable.

A polynomial, where a, b are variable numbers, is called a polynomial of the first degree; a polynomial where a, b, c are variable numbers, called a polynomial of the second degree or a square trinomial; a polynomial where a, b, c, d are numbers, the variable is called a polynomial of the third degree.

In general, if o is a variable, then it is a polynomial

called lsmogochnolenol degree (relative to x); , m-terms of the polynomial, coefficients, the leading term of the polynomial, a is the coefficient of the leading term, the free term of the polynomial. Typically, a polynomial is written in descending powers of a variable, i.e., the powers of a variable gradually decrease, in particular, the leading term is in first place, and the free term is in last place. The degree of a polynomial is the degree of the highest term.

For example, a polynomial of the fifth degree, in which the leading term, 1, is the free term of the polynomial.

The root of a polynomial is the value at which the polynomial vanishes. For example, the number 2 is the root of a polynomial since

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 to convert any polynomial into standard form. Finally, let's look at solutions to 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 polynomial of the standard form.

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 the following rule, which explains 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.

Let us note here 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, 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 bring the monomial 2·a 3 ·0.6 to the standard form, we have 2·a 3 ·0.6=1.2·a 3 , after which – 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 a standard form, this allows us to determine the degree of the polynomial, which is equal to the highest degree of the 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. The term z 5 has the highest degree; 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 .

References.

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. In 2 hours. Part 1. Textbook for students of general education 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.

By definition, a polynomial is algebraic expression which is the sum of monomials.

For example: 2*a^2 + 4*a*x^7 - 3*a*b^3 + 4; 6 + 4*b^3 are polynomials, and the expression z/(x - x*y^2 + 4) is not a polynomial because it is not a sum of monomials. A polynomial is also sometimes called a polynomial, and monomials that are part of a polynomial are members of a polynomial or monomials.

Complex concept of polynomial

If a polynomial consists of two terms, then it is called a binomial; if it consists of three, it is called a trinomial. The names fournomial, fivenomial and others are not used, and in such cases they simply say polynomial. Such names, depending on the number of terms, put everything in its place.

And the term monomial becomes intuitive. From a mathematical point of view, a monomial is a special case of a polynomial. A monomial is a polynomial that consists of one term.

Just like a monomial, a polynomial has its own standard form. The standard form of a polynomial is such a notation of a polynomial in which all the monomials included in it as terms are written in a standard form and similar terms are given.

Standard form of polynomial

The procedure for reducing a polynomial to standard form is to reduce each of the monomials to standard form, and then add all similar monomials together. The addition of similar terms of a polynomial is called reduction of similar.
For example, let's present similar terms in the polynomial 4*a*b^2*c^3 + 6*a*b^2*c^3 - a*b.

The terms 4*a*b^2*c^3 and 6*a*b^2*c^3 are similar here. The sum of these terms will be the monomial 10*a*b^2*c^3. Therefore, the original polynomial 4*a*b^2*c^3 + 6*a*b^2*c^3 - a*b can be rewritten as 10*a*b^2*c^3 - a*b . This entry will be the standard form of a polynomial.

From the fact that any monomial can be reduced to a standard form, it also follows that any polynomial can be reduced to a standard form.

When a polynomial is reduced to standard form, we can talk about such a concept as the degree of a polynomial. The degree of a polynomial is the highest degree of a monomial included in a given polynomial.
So, for example, 1 + 4*x^3 - 5*x^3*y^2 is a polynomial of the fifth degree, since the maximum degree of the monomial included in the polynomial (5*x^3*y^2) is fifth.

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 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.

Next typical task- calculation of a 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: recall that one to any natural power is equal to one, and zero to any natural power is 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 reducing 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 the standard form, you need to bring the first, second and sixth; second action - we bring similar terms, 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 do not have 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 one 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.