Write a polynomial online in standard form. Polynomials

By definition, a polynomial is an algebraic expression representing 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 a 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.

After studying monomials, we move on to polynomials. This article will tell you about all the necessary information required to perform actions on them. We will define a polynomial with accompanying definitions of a polynomial term, that is, free and similar, consider a standard form polynomial, introduce a degree and learn how to find it, and 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 of 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 a special group called similar members of the 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 a 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

Degree of a polynomial of an arbitrary 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 9

When all the 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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Polynomial and its standard form

A polynomial is the sum of monomials.

The monomials that make up a polynomial are called members of the polynomial. So the terms of the polynomial 4x2y - 5xy + 3x -1 are 4x2y, -5xy, 3x and -1.

If a polynomial consists of two terms, then it is called a binomial, if it consists of three, it is called a trinomial. A monomial is considered a polynomial consisting of one term.

In the polynomial 7x3y2 - 12 + 4x2y - 2y2x3 + 6, the terms 7x3y2 and - 2y2x3 are similar terms because they have the same letter part. The terms -12 and 6, which do not have a letter part, are also similar. Similar terms in a polynomial are called similar terms of a polynomial, and the reduction of similar terms in a polynomial is called a reduction of similar terms of a polynomial.

As an example, let us give similar terms in the polynomial 7x3y2 - 12 + 4x2y - 2y2x3 + 6 = 5x3y2 + 4x2y - 6.

A polynomial is called a polynomial of standard form if each of its terms is a monomial of standard form and this polynomial does not contain similar terms.

Any polynomial can be reduced to standard form. To do this, you need to present each of its members in standard form and bring similar terms.

The degree of a polynomial of standard form is the highest of the degrees of its constituent monomials.

The degree of an arbitrary polynomial is the degree of an identically equal polynomial of standard form.

For example, let's find the degree of the polynomial 8x4y2 - 12 + 4x2y - 3y2x4 + 6 - 5y2x4:

8x4y2 - 12 + 4x2y - 3y2x4 + 6 - 5y2x4 = 4x2y -6.

Note that the original polynomial includes monomials of the sixth degree, but when similar terms were reduced, all of them were reduced, and the result was a polynomial of the third degree, which means the original polynomial has degree 3!
Polynomials in one variable

An expression of the form where are some numbers and is called a polynomial of degree from.

Two polynomials are said to be identically equal if they numeric values coincide for all values. Polynomials and are identically equal if and only if they coincide, i.e. the coefficients for the same powers of these polynomials are the same.

When dividing a polynomial by a polynomial (for example, by a “corner”), we obtain a polynomial (incomplete quotient) and a remainder - a polynomial (in the case when the remainder equal to zero, the polynomial is called private). If is the dividend and is the divisor, then we represent the polynomial in the form. In this case, the sum of the degrees of polynomials is equal to the degree of the polynomial, and the degree of the remainder is less than the degree of the divisor.

The concept of a polynomial. Polynomial degree

A polynomial in the variable x is an expression of the form

anxn+an-1xn-1+... +a1x+a0,where n - natural number; аn, an-1,..., a1, a0 - any numbers called the coefficients of this polynomial. The expressions anxn, an-1xn-1,..., a1x, a0 are called terms of the polynomial, a0 is the free term.

We will often use the following terms: an - coefficient for xn, an-1 - coefficient for xn-1, etc.

Examples of polynomials are the following expressions: 0x4+2x3+ (-3) x3+ (3/7) x+; 0x2+0x+3; 0x2+0x+0. Here, for the first polynomial, the coefficients are the numbers 0, 2, - 3, 3/7, ; in this case, for example, the number 2 is the coefficient of x3, and is the free term.

A polynomial whose coefficients are all zero is called zero.

So, for example, the polynomial 0x2+0x+0 is zero.

From the notation of a polynomial it is clear that it consists of several members. This is where the term ‹‹polynomial›› (many terms) comes from. Sometimes a polynomial is called a polynomial. This term comes from the Greek words πολι - many and νομχ - member.

We will denote a polynomial in one variable x as follows: f (x), g (x), h (x), etc. for example, if the first of the above polynomials is denoted by f (x), then we can write: f (x) =0x4+2x3+ (-3) x2+3/7x+.

In order to make the polynomial notation simpler and more compact, we agreed on a number of conventions.

Those terms of a non-zero polynomial whose coefficients are equal to zero are not written down. For example, instead of f (x) =0x3+3x2+0x+5 they write: f (x) =3x2+5; instead of g (x) =0x2+0x+3 - g (x) =3. Thus, every number is also a polynomial. A polynomial h (x) for which all coefficients are equal to zero, i.e. zero polynomial is written as follows: h (x) =0.

Coefficients of a polynomial that are not a free term and equal to 1 are also not written down. For example, the polynomial f (x) =2x3+1x2+7x+1 can be written as follows: f (x) =x3+x2+7x+1.

The sign ‹‹-›› of a negative coefficient is assigned to the term containing this coefficient, i.e., for example, the polynomial f (x) =2x3+ (-3) x2+7x+ (-5) is written as f (x) =2x3 -3x2+7x-5. Moreover, if the coefficient, which is not a free term, is equal to - 1, then the “-” sign is kept in front of the corresponding term, and the unit is not written. For example, if the polynomial has the form f (x) =x3+ (-1) x2+3x+ (-1), then it can be written as follows: f (x) =x3-x2+3x-1.

The question may arise: why, for example, agree to replace 1x with x in the notation of a polynomial if it is known that 1x = x for any number x? The point is that the last equality holds if x is a number. In our case, x is an element of arbitrary nature. Moreover, we do not yet have the right to consider the entry 1x as the product of the number 1 and the element x, because, we repeat, x is not a number. It is precisely this circumstance that causes the conventions in writing a polynomial. And if we continue to talk about the product of, say, 2 and x without any reason, then we are admitting some lack of rigor.

Due to conventions in writing a polynomial, we pay attention to this detail. If there is, for example, a polynomial f (x) = 3x3-2x2-x+2, then its coefficients are the numbers 3, - 2, - 1.2. Of course, one could say that the coefficients are the numbers 0, 3, - 2, - 1, 2, meaning this representation of this polynomial: f (x) = 0x4-3x2-2x2-x+2.

In the future, for definiteness, we will indicate the coefficients, starting with non-zero ones, in the order they appear in the notation of the polynomial. Thus, the coefficients of the polynomial f (x) = 2x5-x are the numbers 2, 0, 0, 0, - 1, 0. The fact is that although, for example, the term with x2 is absent in the notation, this only means that its coefficient equal to zero. Similarly, there is no free term in the entry, since it is equal to zero.

If there is a polynomial f (x) =anxn+an-1xn-1+... +a1x+a0 and an≠0, then the number n is called the degree of the polynomial f (x) (or they say: f (x) - nth degree) and write Art. f(x)=n. In this case, an is called the leading coefficient, and anxn is the leading term of this polynomial.

For example, if f (x) =5x4-2x+3, then art. f (x) =4, leading coefficient - 5, leading term - 5x4.

Let us now consider the polynomial f (x) =a, where a is a non-zero number. What is the degree of this polynomial? It is easy to see that the coefficients of the polynomial f (x) =anxn+an-1xn-1+... +a1x+a0 are numbered from right to left with the numbers 0, 1, 2, …, n-1, n and if an≠0, then Art. f(x)=n. This means that the degree of a polynomial is the largest of the numbers of its coefficients that are different from zero (with the numbering that was just mentioned). Let us now return to the polynomial f (x) =a, a≠0, and number its coefficients from right to left with the numbers 0, 1, 2, ... coefficient a will receive the number 0, and since all other coefficients are zero, then this is the largest non-zero coefficient number of a given polynomial. So art. f (x) =0.

Thus, polynomials of degree zero are numbers other than zero.

It remains to find out what the situation is with the degree of the zero polynomial. As is known, all its coefficients are equal to zero, and therefore the above definition cannot be applied to it. So, we agreed not to assign any degree to the zero polynomial, i.e. that he doesn't have a degree. This convention is caused by some circumstances that will be discussed a little later.

So, the zero polynomial has no degree; the polynomial f (x) =a, where a is a non-zero number and has degree 0; the degree of any other polynomial, as is easy to see, is equal to the largest exponent of the variable x, the coefficient of which is equal to zero.

In conclusion, let us recall a few more definitions. A polynomial of the second degree f (x) =ax2+bx+c is called a square trinomial. A polynomial of the first degree of the form g (x) =x+c is called a linear binomial.
Horner's scheme.

Horner's scheme is one of the simplest ways to divide a polynomial by a binomial x-a. Of course, the application of Horner’s scheme is not limited to division, but first let’s consider just that. We will explain the use of the algorithm with examples. Divide by. Let's make a table of two lines: in the first line we write the coefficients of the polynomial in descending order of degrees of the variable. Note that this polynomial does not contain x, i.e. the coefficient in front of x is 0. Since we are dividing by, we write one in the second line:

Let's start filling in the empty cells in the second line. Let's write 5 into the first empty cell, simply moving it from the corresponding cell of the first row:

Let's fill the next cell according to this principle:

Let's fill in the fourth one in the same way:

For the fifth cell we get:

And finally, for the last, sixth cell, we have:

The problem is solved, all that remains is to write down the answer:

As you can see, the numbers located in the second line (between the first and last) are the coefficients of the polynomial obtained after dividing by. Last number in the second line means the remainder of the division or, which is the same, the value of the polynomial at. Consequently, if in our case the remainder is equal to zero, then the polynomials are divided entirely.

The result also indicates that 1 is the root of the polynomial.

Let's give another example. Let's divide the polynomial by. Let us immediately stipulate that the expression must be presented in the form. Horner’s scheme will involve exactly -3.

If our goal is to find all the roots of a polynomial, then Horner's scheme can be applied several times in a row until we have exhausted all the roots. For example, let's find all the roots of a polynomial. Whole roots must be looked for among the divisors of the free term, i.e. among the divisors there are 8. That is, the numbers -8, -4, -2, -1, 1, 2, 4, 8 can be integer roots. Let's check, for example, 1:

So, the remainder is 0, i.e. unity is indeed the root of this polynomial. Let's try to check the unit a few more times. We will not create a new table for this, but will continue to use the previous one:

Again the remainder is zero. Let's continue the table until we have exhausted all possible root values:

Bottom line: Of course this method selection is ineffective in the general case, when the roots are not integers, but for integer roots the method is quite good.

RATIONAL ROOTS OF A POLYNOMIAL WITH INTEGER COEFFICIENTS Finding the roots of a polynomial is an interesting and rather difficult problem, the solution of which goes beyond the limits of school course mathematics. However, for polynomials with integer coefficients there is a simple enumeration algorithm that allows you to find all rational roots.

Theorem. If a polynomial with integer coefficients has a rational root (is an irreducible fraction),

then the numerator of the fraction is the divisor of the free term, and the denominator is the divisor of the leading coefficient of this polynomial.

Proof

Let the polynomial be written in canonical form. Let us substitute and get rid of the denominators by multiplying by the largest power n:

Move the member to the right

The product is divided by the integer m. By condition, the fraction is irreducible, therefore, the numbers m and n are coprime. Then the numbers m will be coprime and If the product of numbers is divisible by m, and the factor is coprime by m, then the second factor must be divisible by m.

The proof of the divisibility of the leading coefficient by the denominator n is proved in the same way, moving the term to the right and moving the factor n out of the left bracket from the left.

Let us make a few comments to the proven theorem.

Notes

1) The theorem gives only necessary condition existence of a rational root. This means that you need to check all rational numbers with the property specified in the theorem and select from them those that turn out to be roots. There will be no others.

2) Among the divisors, you must take not only positive, but also negative integers.

3) If the leading coefficient is 1, then every rational root must be an integer, since 1 has no divisors except

Let us illustrate the theorem and comments to it with examples.

1) Rational roots must be whole.

We sort out the divisors of the free term: There is no point in substituting positive numbers, since all the coefficients of the polynomial are positive and at

It remains to calculate F(–1) and F(–2). F(–1)=1+0; F(–2)=0.

So the polynomial has one whole root x=–2.

We can divide F(x) by x+2:

2) Write down the possible values ​​of the roots:

By substitution we are convinced that the Polynomial also has three different rational roots:

Of course, the root x = -1 is easy to guess. Then you can factorize and look for the roots of the quadratic trinomial using the usual techniques.

DIVISION OF POLYNOMIALS. EUCLID ALGORITHM

Division of polynomials

The result of division is a single pair of polynomials - the quotient and the remainder, which must satisfy the equality:< делимое > = < делитель > ´ < частное > + <… Если многочлен степени n Pn(x) является делимым,

Example No. 1

6x 3 + x 2 – 3x – 2 2x 2 – x – 1

6x 3 ± 3x 2 ± 3x 3x + 2

4x 2 + 0x – 2

4x 2 ± 2x ± 2

Thus, 6x 3 + x 2 – 3x – 2 = (2x 2 – x – 1)(3x + 2) + 2x.

Example No. 2

a 5 a 4 b a 4 –a 3 b + a 2 b 2 – ab 3 + b 4

± a 4 b ± a 3 b 2

– a 2 b 3 + b 5

± a 2 b 3 ± ab 4

Thus, a 5 + b 5 = (a + b)(a 4 –a 3 b + a 2 b 2 – ab 3 + b 4).

- polynomials. In this article we will outline all the initial and necessary information about polynomials. These include, firstly, the definition of a polynomial with accompanying definitions of the terms of the polynomial, in particular, the free term and similar terms. Secondly, we will dwell on polynomials of the standard form, give the corresponding definition and give examples of them. Finally, we will introduce the definition of the degree of a polynomial, figure out how to find it, and talk about the coefficients of the terms of the polynomial.

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Polynomial and its terms - definitions and examples

In grade 7, polynomials are studied immediately after monomials, this is understandable, since polynomial definition is given through monomials. Let us give this definition to explain what a polynomial is.

Definition.

Polynomial is the sum of monomials; A monomial is considered a special case of a polynomial.

The written definition allows you to give as many examples of polynomials as you like. Any of the monomials 5, 0, −1, x, 5 a b 3, x 2 0.6 x (−2) y 12, etc. is a polynomial. Also, by definition, 1+x, a 2 +b 2 and are polynomials.

For the convenience of describing polynomials, a definition of a polynomial term is introduced.

Definition.

Polynomial terms are the constituent monomials of a polynomial.

For example, the polynomial 3 x 4 −2 x y+3−y 3 consists of four terms: 3 x 4 , −2 x y , 3 and −y 3 . A monomial is considered a polynomial consisting of one term.

Definition.

Polynomials that consist of two and three terms have special names - binomial And trinomial respectively.

So x+y is a binomial, and 2 x 3 q−q x x x+7 b is a trinomial.

At school, we most often have to work with linear binomial a x+b , where a and b are some numbers, and x is a variable, as well as c quadratic trinomial a·x 2 +b·x+c, where a, b and c are some numbers, and x is a variable. Here are examples of linear binomials: x+1, x 7,2−4, and here are examples of square trinomials: x 2 +3 x−5 and .

Polynomials in their notation can have similar terms. For example, in the polynomial 1+5 x−3+y+2 x the similar terms are 1 and −3, as well as 5 x and 2 x. They have their own special name - similar terms of a polynomial.

Definition.

Similar terms of a polynomial similar terms in a polynomial are called.

In the previous example, 1 and −3, as well as the pair 5 x and 2 x, are similar terms of the polynomial. In polynomials that have similar terms, you can reduce similar terms to simplify their form.

Polynomial of standard form

For polynomials, as for monomials, there is a so-called standard form. Let us voice the corresponding definition.

Based on this definition, we can give examples of polynomials of the standard form. So the polynomials 3 x 2 −x y+1 and written in standard form. And the expressions 5+3 x 2 −x 2 +2 x z and x+x y 3 x z 2 +3 z are not polynomials of the standard form, since the first of them contains similar terms 3 x 2 and −x 2 , and in the second – a monomial x·y 3 ·x·z 2 , the form of which is different from the standard one.

Note that, if necessary, you can always reduce the polynomial to standard form.

Another concept related to polynomials of the standard form is the concept of a free term of a polynomial.

Definition.

Free term of a polynomial is a member of a polynomial of standard form without a letter part.

In other words, if a polynomial of standard form contains a number, then it is called a free member. For example, 5 is the free term of the polynomial x 2 z+5, but the polynomial 7 a+4 a b+b 3 does not have a free term.

Degree of a polynomial - how to find it?

Another important related definition is the definition of the degree of a polynomial. First, we define the degree of a polynomial of the standard form; this definition is based on the degrees of the monomials that are in its composition.

Definition.

Degree of a polynomial of standard form is the largest of the powers of the monomials included in its notation.

Let's give examples. The degree of the polynomial 5 x 3 −4 is equal to 3, since the monomials 5 x 3 and −4 included in it have degrees 3 and 0, respectively, the largest of these numbers is 3, which is the degree of the polynomial by definition. And the degree of the polynomial 4 x 2 y 3 −5 x 4 y+6 x equal to the largest of the numbers 2+3=5, 4+1=5 and 1, that is, 5.

Now let's find out how to find the degree of a polynomial of any form.

Definition.

The degree of a polynomial of arbitrary form call the degree of the corresponding polynomial of standard form.

So, if a polynomial is not written in standard form, and you need to find its degree, then you need to reduce the original polynomial to standard form, and find the degree of the resulting polynomial - it will be the required one. Let's look at the example solution.

Example.

Find the degree of the polynomial 3 a 12 −2 a b c a c b+y 2 z 2 −2 a 12 −a 12.

Solution.

First you need to represent the polynomial in standard form:
3 a 12 −2 a b 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.

The resulting polynomial of standard form includes two monomials −2·a 2 ·b 2 ·c 2 and y 2 ·z 2 . Let's find their powers: 2+2+2=6 and 2+2=4. Obviously, the largest of these powers is 6, which by definition is the power of a polynomial of the standard form −2 a 2 b 2 c 2 +y 2 z 2, and therefore the degree of the original polynomial., 3 x and 7 of the polynomial 2 x−0.5 x y+3 x+7 .

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. 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 the beginning of 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.

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

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 the 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 absolute common divisor of all coefficients of the polynomial is taken as the coefficient of the common factor.

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: Factor .

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 do the grouping 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