Definition.The sum of integer non-negative powers of an unknown X, taken with certain numerical coefficients, is called a polynomial.
Here: 
- real numbers.
n- degree of the polynomial.
Operations on polynomials.
1). When adding (subtracting) two polynomials, the coefficients are added (subtracted) equal degrees unknown x.
2). Two polynomials are equal if they have the same degree and equal coefficients at the same powers of X.
3). The degree of a polynomial obtained by multiplying two polynomials is equal to the sum of the degrees of the polynomials being multiplied.
4). Linear operations on polynomials have the properties of associativity, commutativity and distributivity.
5) The division of a polynomial by a polynomial can be done using the “division by a corner” rule.
Definition. The number x=a is called the root of a polynomial if its substitution into a polynomial turns it into zero, i.e.
Bezout's theorem. Polynomial remainder
by binomial (x-a) is equal to the value of the polynomial at x=a, i.e.
Proof.
Let where
Putting x=a in the equality, we get
1). When dividing a polynomial by a binomial (x-a), the remainder will always be a number.
2). If a is the root of a polynomial, then the polynomial is divisible by the binomial (x-a) without a remainder.
3) When dividing a polynomial of degree n by a binomial (x-a), we obtain a polynomial of degree (n-1).
Fundamental theorem of algebra.Any polynomial of degreen (n>1) has at least one root(presented without proof).
Consequence.Any polynomial of degree
n
has exactly
n
roots and over the field of complex numbers is decomposed into the product
n
linear factors, i.e.
Among the roots of the polynomial 
there may be repeating numbers (multiple roots). For polynomials with real coefficients, complex roots can appear only in conjugate pairs. Let us prove the last statement.
Let 
- complex root polynomial, then Based on general property complex numbers can be stated therefore 
- also a root.
Each pair of complex conjugate roots of a polynomial corresponds to a square trinomial with real coefficients.
Here p, q- real numbers (show example).
Conclusion.We can represent any polynomial as a product of linear factors and square trinomials with real coefficients.
Rational fractions.
A rational fraction is the ratio of two polynomials.
If 
, then the rational fraction is called proper. IN otherwise the fraction is incorrect. Any improper fraction can be represented as the sum of a polynomial (quotient) and a proper rational fraction by dividing the polynomial in the numerator by the polynomial in the denominator.
- improper rational fraction.
This improper rational fraction can now be represented in the following form.
Taking into account what has been shown, in the future we will consider only proper rational fractions.
There are so-called simple rational fractions - these are fractions that cannot be simplified in any way. These simplest fractions look like:
A proper rational fraction of a more complex form can always be represented as a sum of the simplest rational fractions. The set of fractions is determined by the set of roots of the polynomial that appears in the denominator of a proper irreducible rational fraction. The rule for decomposing a fraction into its simplest is as follows.
Let the rational fraction be represented in the following form.
Here, the numerator of the simplest fractions contains unknown coefficients, which can always be determined by the method uncertain coefficients. The essence of the method is to equate the coefficients at the same powers of X for the polynomial in the numerator of the original fraction and the polynomial in the numerator of the fraction obtained after reducing the simplest fractions to a common denominator.
Let us equate the coefficients for the same powers of X.
Solving the system of equations for unknown coefficients, we obtain.
So, given fraction can be represented by a set of the following simple fractions.
Leading to common denominator We make sure that the problem is solved correctly.
Any fractional expression (clause 48) can be written in the form , where P and Q are rational expressions, and Q necessarily contains variables. Such a fraction is called a rational fraction.
Examples of rational fractions:
The main property of a fraction is expressed by an identity that is fair under the conditions here - a whole rational expression. This means that the numerator and denominator of a rational fraction can be multiplied or divided by the same non-zero number, monomial or polynomial.
For example, the property of a fraction can be used to change the signs of members of a fraction. If the numerator and denominator of a fraction are multiplied by -1, we get Thus, the value of the fraction will not change if the signs of the numerator and denominator are simultaneously changed. If you change the sign of only the numerator or only the denominator, then the fraction will change its sign:
For example,
60. Reducing rational fractions.
To reduce a fraction means to divide the numerator and denominator of the fraction by a common factor. The possibility of such a reduction is due to the basic property of the fraction.
In order to reduce a rational fraction, you need to factor the numerator and denominator. If it turns out that the numerator and denominator have common factors, then the fraction can be reduced. If there are no common factors, then converting a fraction through reduction is impossible.
Example. Reduce fraction
Solution. We have
The reduction of a fraction is carried out under the condition .
61. Reducing rational fractions to a common denominator.
The common denominator of several rational fractions is a whole rational expression that is divided by the denominator of each fraction (see paragraph 54).
For example, the common denominator of fractions is a polynomial since it is divisible by both and by and polynomial and polynomial and polynomial, etc. Usually they take such a common denominator that any other common denominator is divisible by Echosen. Such simplest denominator sometimes called the lowest common denominator.
In the example discussed above, the common denominator is We have
Reducing these fractions to a common denominator is achieved by multiplying the numerator and denominator of the first fraction by 2. and the numerator and denominator of the second fraction by Polynomials are called additional factors for the first and second fractions, respectively. The additional factor for a given fraction is equal to the quotient of dividing the common denominator by the denominator of the given fraction.
To reduce several rational fractions to a common denominator, you need:
1) factor the denominator of each fraction;
2) create a common denominator by including as factors all the factors obtained in step 1) of the expansions; if a certain factor is present in several expansions, then it is taken with an exponent equal to the largest of the available ones;
3) find additional factors for each of the fractions (for this, the common denominator is divided by the denominator of the fraction);
4) by multiplying the numerator and denominator of each fraction by an additional factor, bring the fraction to a common denominator.
Example. Reduce a fraction to a common denominator
Solution. Let's factorize the denominators:
The following factors must be included in the common denominator: and the least common multiple of the numbers 12, 18, 24, i.e. This means that the common denominator has the form
Additional factors: for the first fraction for the second for the third. So, we get:
62. Addition and subtraction of rational fractions.
The sum of two (and in general any finite number) rational fractions with same denominators is identically equal to a fraction with the same denominator and numerator, equal to the amount numerators of added fractions:
The situation is similar in the case of subtracting fractions with like denominators:
Example 1: Simplify an expression
Solution.
To add or subtract rational fractions with different denominators You must first reduce the fractions to a common denominator, and then perform operations on the resulting fractions with the same denominators.
Example 2: Simplify an expression
Solution. We have
63. Multiplication and division of rational fractions.
The product of two (and in general any finite number) rational fractions is identically equal to the fraction whose numerator equal to the product numerators, and the denominator - the product of the denominators of the multiplied fractions:
The quotient of dividing two rational fractions is identically equal to a fraction whose numerator is equal to the product of the numerator of the first fraction and the denominator of the second fraction, and the denominator is the product of the denominator of the first fraction and the numerator of the second fraction:
The formulated rules of multiplication and division also apply to the case of multiplication or division by a polynomial: it is enough to write this polynomial in the form of a fraction with a denominator of 1.
Given the possibility of reducing a rational fraction obtained as a result of multiplying or dividing rational fractions, they usually strive to factorize the numerators and denominators of the original fractions before performing these operations.
Example 1: Perform multiplication
Solution. We have
Using the rule for multiplying fractions, we get:
Example 2: Perform division
Solution. We have
Using the division rule, we get:
64. Raising a rational fraction to a whole power.
To raise a rational fraction - to natural degree, you need to raise the numerator and denominator of the fraction to this power separately; the first expression is the numerator, and the second expression is the denominator of the result:
Example 1: Convert to a fraction of power 3.
Solution Solution.
When raising a fraction to a whole number negative degree an identity is used that is valid for all values of the variables for which .
Example 2: Convert an expression to a fraction
65. Transformation of rational expressions.
Transforming any rational expression comes down to adding, subtracting, multiplying and dividing rational fractions, as well as raising a fraction to a natural power. Any rational expression can be converted into a fraction, the numerator and denominator of which are whole rational expressions; this, as a rule, is the goal of identity transformations rational expressions.
Example. Simplify an expression
66. The simplest transformations of arithmetic roots (radicals).
When converting arithmetic korias, their properties are used (see paragraph 35).
Let's look at a few examples of using properties arithmetic roots for the simplest transformations of radicals. In this case, we will consider all variables to take only non-negative values.
Example 1. Extract the root of a product
Solution. Applying the 1° property, we get:
Example 2. Remove the multiplier from under the root sign
Solution.
This transformation is called removing the factor from under the root sign. The purpose of the transformation is to simplify the radical expression.
Example 3: Simplify.
Solution. By the property of 3° we have. Usually they try to simplify the radical expression, for which they take the factors out of the corium sign. We have
Example 4: Simplify
Solution. Let's transform the expression by introducing a factor under the sign of the root: By property 4° we have
Example 5: Simplify
Solution. By the property of 5°, we have the right to divide the exponent of the root and the exponent of the radical expression into the same thing natural number. If in the example under consideration we divide the indicated indicators by 3, we get .
Example 6. Simplify expressions:
Solution, a) By property 1° we find that to multiply roots of the same degree, it is enough to multiply the radical expressions and extract the root of the same degree from the result obtained. Means,
b) First of all, we must reduce the radicals to one indicator. According to the property of 5°, we can multiply the exponent of the root and the exponent of the radical expression by the same natural number. Therefore, Next, we now have in the resulting result dividing the exponents of the root and the degree of the radical expression by 3, we get.
Let's start with some definitions. Polynomial nth degree(or nth order) we will call an expression of the form $P_n(x)=\sum\limits_(i=0)^(n)a_(i)x^(n-i)=a_(0)x^(n)+ a_(1)x^(n-1)+a_(2)x^(n-2)+\ldots+a_(n-1)x+a_n$. For example, the expression $4x^(14)+87x^2+4x-11$ is a polynomial whose degree is $14$. It can be denoted as follows: $P_(14)(x)=4x^(14)+87x^2+4x-11$.
The ratio of two polynomials $\frac(P_n(x))(Q_m(x))$ is called rational function or rational fraction. To be more precise, this is rational function one variable (i.e. variable $x$).
The rational fraction is called correct, if $n< m$, т.е. если степень многочлена, стоящего в числителе, less degree polynomial in the denominator. Otherwise (if $n ≥ m$) the fraction is called wrong.
Example No. 1
Indicate which of the following fractions are rational. If the fraction is rational, then find out whether it is correct or not.
- $\frac(3x^2+5\sin x-4)(2x+5)$;
- $\frac(5x^2+3x-8)(11x^9+25x^2-4)$;
- $\frac((2x^3+8x+4)(8x^4+5x^3+x+145)^9(5x^7+x^6+9x^5+3))((5x+4) (3x^2+9)^(15)(15x^(10)+9x-1))$;
- $\frac(3)((5x^6+4x+19)^4)$.
1) This fraction is not rational because it contains $\sin x$. A rational fraction does not allow this.
2) We have the ratio of two polynomials: $5x^2+3x-8$ and $11x^9+25x^2-4$. Therefore, according to the definition, the expression $\frac(5x^2+3x-8)(11x^9+25x^2-4)$ is a rational fraction. Since the degree of the polynomial in the numerator is equal to $2$, and the degree of the polynomial in the denominator is equal to $9$, then this fraction is proper (since $2< 9$).
3) Both the numerator and the denominator of this fraction contain polynomials (factored). It doesn’t matter to us at all in what form the numerator and denominator polynomials are presented: whether they are factorized or not. Since we have a ratio of two polynomials, then according to the definition the expression $\frac((2x^3+8x+4)(8x^4+5x^3+x+145)^9(5x^7+x^6+9x ^5+3))((5x+4)(3x^2+9)^(15)(15x^(10)+9x-1))$ is a rational fraction.
In order to answer the question of whether a given fraction is proper, one must determine the powers of the polynomials in the numerator and denominator. Let's start with the numerator, i.e. from the expression $(2x^3+8x+4)(8x^4+5x^3+x+145)^9(5x^7+x^6+9x^5+3)$. To determine the degree of this polynomial, you can, of course, open the brackets. However, it is much simpler to act rationally, because we are only interested in greatest degree variable $x$. From each bracket we choose the variable $x$ to the greatest degree. From the bracket $(2x^3+8x+4)$ we take $x^3$, from the bracket $(8x^4+5x^3+x+9)^9$ we take $(x^4)^9=x ^(4\cdot9)=x^(36)$, and from the bracket $(5x^7+x^6+9x^5+3)$ we choose $x^7$. Then, after opening the parentheses, the largest power of the variable $x$ will be like this:
$$ x^3\cdot x^(36)\cdot x^7=x^(3+36+7)=x^(46). $$
The degree of the polynomial located in the numerator is $46$. Now let's turn to the denominator, i.e. to the expression $(5x+4)(3x^2+9)^(15)(15x^(10)+9x-1)$. The degree of this polynomial is determined in the same way as for the numerator, i.e.
$$ x\cdot (x^2)^(15)\cdot x^(10)=x^(1+30+10)=x^(41). $$
The denominator contains a polynomial of degree 41. Since the degree of the polynomial in the numerator (i.e. 46) is not less than the degree of the polynomial in the denominator (i.e. 41), then the rational fraction is $\frac((2x^3+8x+4)(8x^4+5x^ 3+x+145)^9(5x^7+x^6+9x^5+3))((5x+4)(3x^2+9)^(15)(15x^(10)+9x- 1))$ is incorrect.
4) The numerator of the fraction $\frac(3)((5x^6+4x+19)^4)$ contains the number $3$, i.e. polynomial zero degree. Formally, the numerator can be written as follows: $3x^0=3\cdot1=3$. In the denominator we have a polynomial whose degree is equal to $6\cdot 4=24$. The ratio of two polynomials is a rational fraction. Since $0< 24$, то данная дробь является правильной.
Answer: 1) the fraction is not rational; 2) rational fraction (proper); 3) rational fraction (irregular); 4) rational fraction (proper).
Now let's move on to the concept of elementary fractions (they are also called the simplest rational fractions). There are four types of elementary rational fractions:
- $\frac(A)(x-a)$;
- $\frac(A)((x-a)^n)$ ($n=2,3,4,\ldots$);
- $\frac(Mx+N)(x^2+px+q)$ ($p^2-4q< 0$);
- $\frac(Mx+N)((x^2+px+q)^n)$ ($p^2-4q< 0$; $n=2,3,4,\ldots$).
Note (desirable for a more complete understanding of the text): show\hide
Why is the condition $p^2-4q needed?< 0$ в дробях третьего и четвертого типов? Рассмотрим quadratic equation$x^2+px+q=0$. The discriminant of this equation is $D=p^2-4q$. Essentially, the condition $p^2-4q< 0$ означает, что $D < 0$. Если $D < 0$, то уравнение $x^2+px+q=0$ не имеет real roots. Those. the expression $x^2+px+q$ cannot be factorized. It is this indecomposability that interests us.
For example, for the expression $x^2+5x+10$ we get: $p^2-4q=5^2-4\cdot 10=-15$. Since $p^2-4q=-15< 0$, то выражение $x^2+5x+10$ нельзя разложить на множители.
By the way, for this check it is not at all necessary that the coefficient before $x^2$ be equal to 1. For example, for $5x^2+7x-3=0$ we get: $D=7^2-4\cdot 5 \cdot (-3)=$109. Since $D > 0$, the expression $5x^2+7x-3$ is factorizable.
The task is as follows: given correct represent a rational fraction as a sum of elementary rational fractions. The material presented on this page is devoted to solving this problem. First you need to make sure that you have completed next condition: the polynomial in the denominator of a proper rational fraction is factorized in such a way that this expansion contains only brackets of the form $(x-a)^n$ or $(x^2+px+q)^n$ ($p^2-4q< 0$).Грубо говоря, это требование означает необходимость максимального разложения многочлена в знаменателе, т.е. чтобы дальнейшее разложение было невозможно. Только если это условие выполнено, то можно применять такую схему:
- Each bracket of the form $(x-a)$ located in the denominator corresponds to a fraction $\frac(A)(x-a)$.
- Each bracket of the form $(x-a)^n$ ($n=2,3,4,\ldots$) located in the denominator corresponds to a sum of $n$ fractions: $\frac(A_1)(x-a)+\frac( A_2)((x-a)^2)+\frac(A_3)((x-a)^3)+\ldots+\frac(A_n)((x-a)^n)$.
- Each bracket of the form $(x^2+px+q)$ ($p^2-4q< 0$), расположенной в знаменателе, соответствует дробь $\frac{Cx+D}{x^2+px+q}$.
- Each bracket of the form $(x^2+px+q)^n$ ($p^2-4q< 0$; $n=2,3,4,\ldots$), расположенной в знаменателе, соответствует сумма из $n$ дробей: $\frac{C_1x+D_1}{x^2+px+q}+\frac{C_2x+D_2}{(x^2+px+q)^2}+\frac{C_3x+D_3}{(x^2+px+q)^3}+\ldots+\frac{C_nx+D_n}{(x^2+px+q)^n}$.
If the fraction is improper, then before applying the above scheme, you should divide it into the sum of the integer part (polynomial) and the proper rational fraction. We’ll look at how exactly this is done further (see example No. 2, point 3). A few words about letter designations in numerators (i.e. $A$, $A_1$, $C_2$ and the like). You can use any letters to suit your taste. It is only important that these letters be various in all elementary fractions. To find the values of these parameters, use the method of undetermined coefficients or the method of substituting partial values (see examples No. 3, No. 4 and No. 5).
Example No. 2
Decompose the given rational fractions into elementary ones (without finding the parameters):
- $\frac(5x^4-10x^3+x^2-9)((x-5)(x+2)^4 (x^2+3x+10)(x^2+11)^5) $;
- $\frac(x^2+10)((x-2)^3(x^3-8)(3x+5)(3x^2-x-10))$;
- $\frac(3x^5-5x^4+10x^3-16x^2-7x+22)(x^3-2x^2+4x-8)$.
1) We have a rational fraction. The numerator of this fraction contains a polynomial of degree 4, and the denominator contains a polynomial whose degree is equal to $17$ (how to determine this degree is explained in detail in paragraph No. 3 of example No. 1). Since the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, this fraction is proper. Let's turn to the denominator of this fraction. Let's start with the brackets $(x-5)$ and $(x+2)^4$, which completely fall under the form $(x-a)^n$. In addition, there are also brackets $(x^2+3x+10)$ and $(x^2+11)^5$. The expression $(x^2+3x+10)$ has the form $(x^2+px+q)^n$, where $p=3$; $q=10$, $n=1$. Since $p^2-4q=9-40=-31< 0$, то данную скобку больше нельзя разложить на множители. Обратимся ко второй скобке, т.е. $(x^2+11)^5$. Это тоже скобка вида $(x^2+px+q)^n$, но на сей раз $p=0$, $q=11$, $n=5$. Так как $p^2-4q=0-121=-121 < 0$, то данную скобку больше нельзя разложить на множители. Итак, мы имеем next output: the polynomial in the denominator is factorized in such a way that this factorization contains only brackets of the form $(x-a)^n$ or $(x^2+px+q)^n$ ($p^2-4q< 0$). Теперь можно переходить и к элементарным дробям. Мы будем применять правила , изложенные выше. Согласно правилу скобке $(x-5)$ будет соответствовать дробь $\frac{A}{x-5}$. Это можно записать так:
$$ \frac(5x^4-10x^3+x^2-9)((x-5)(x+2)^4 (x^2+3x+10)(x^2+11)^5 )=\frac(A)(x-5)+\ldots $$
The result can be written as follows:
$$ 3x^5-5x^4+10x^3-16x^2-7x+22=(x^3-2x^2+4x-8)(3x^2+x)+4x^2+x+22 . $$
Then the fraction $\frac(3x^5-5x^4+10x^3-16x^2-7x+22)(x^3-2x^2+4x-8)$ can be represented in another form:
$$ \frac(3x^5-5x^4+10x^3-16x^2-7x+22)(x^3-2x^2+4x-8)=\frac((x^3-2x^2 +4x-8)(3x^2+x)+4x^2+x+22)(x^3-2x^2+4x-8)=\\ =\frac((x^3-2x^2+ 4x-8)(3x^2+x))(x^3-2x^2+4x-8)+\frac(4x^2+x+22)(x^3-2x^2+4x-8) =\\ =3x^2+x+\frac(4x^2+x+22)(x^3-2x^2+4x-8). $$
The fraction $\frac(4x^2+x+22)(x^3-2x^2+4x-8)$ is a proper rational fraction, because the degree of the polynomial in the numerator (i.e. 2) is less than the degree of the polynomial in the denominator ( i.e. 3). Now let's look at the denominator of this fraction. The denominator contains a polynomial that needs to be factorized. Sometimes Horner's scheme is useful for factorization, but in our case it is easier to get by with the standard “school” method of grouping terms:
$$ x^3-2x^2+4x-8=x^2\cdot(x-2)+4\cdot(x-2)=(x-2)\cdot(x^2+4);\ \ 3x^2+x+\frac(4x^2+x+22)(x^3-2x^2+4x-8)=3x^2+x+\frac(4x^2+x+22)((x -2)\cdot(x^2+4)) $$
Using the same methods as in previous paragraphs, we get:
$$ \frac(4x^2+x+22)((x-2)\cdot(x^2+4))=\frac(A)(x-2)+\frac(Cx+D)(x ^2+4) $$
So, finally we have:
$$ \frac(3x^5-5x^4+10x^3-16x^2-7x+22)(x^3-2x^2+4x-8)=3x^2+x+\frac(A)( x-2)+\frac(Cx+D)(x^2+4) $$
This topic will be continued in the second part.
From the algebra course school curriculum Let's get down to specifics. In this article we will study in detail special kind rational expressions – rational fractions, and also consider what characteristic identical conversions of rational fractions take place.
Let us immediately note that rational fractions in the sense in which we define them below are called algebraic fractions in some algebra textbooks. That is, in this article we will understand rational and algebraic fractions as the same thing.
As usual, let's start with a definition and examples. Next we’ll talk about bringing a rational fraction to a new denominator and changing the signs of the members of the fraction. After this, we will look at how to reduce fractions. Finally, let's look at representing a rational fraction as a sum of several fractions. We will provide all information with examples detailed descriptions decisions.
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Definition and examples of rational fractions
Rational fractions are studied in algebra lessons in 8th grade. We will use the definition of a rational fraction, which is given in the algebra textbook for 8th grade by Yu. N. Makarychev et al.
IN this definition it is not specified whether the polynomials in the numerator and denominator of a rational fraction must be polynomials standard view or not. Therefore, we will assume that the notations for rational fractions can contain both standard and non-standard polynomials.
Here are a few examples of rational fractions. So, x/8 and 
- rational fractions. And fractions 
and do not fit the stated definition of a rational fraction, since in the first of them the numerator does not contain a polynomial, and in the second, both the numerator and the denominator contain expressions that are not polynomials.
Converting the numerator and denominator of a rational fraction
The numerator and denominator of any fraction are self-sufficient mathematical expressions, in the case of rational fractions, these are polynomials; in a particular case, monomials and numbers. Therefore, identical transformations can be carried out with the numerator and denominator of a rational fraction, as with any expression. In other words, the expression in the numerator of a rational fraction can be replaced by an identically equal expression, just like the denominator.
You can perform identical transformations in the numerator and denominator of a rational fraction. For example, in the numerator you can group and reduce similar terms, and in the denominator, replace the product of several numbers with its value. And since the numerator and denominator of a rational fraction are polynomials, it is possible to perform transformations characteristic of polynomials with them, for example, reduction to a standard form or representation in the form of a product.
For clarity, let's consider solutions to several examples.
Example.
Convert rational fraction 
so that the numerator contains a polynomial of standard form, and the denominator contains the product of polynomials.
Solution.
Reducing rational fractions to a new denominator is mainly used in adding and subtracting rational fractions.
Changing signs in front of a fraction, as well as in its numerator and denominator
The main property of a fraction can be used to change the signs of the members of a fraction. Indeed, multiplying the numerator and denominator of a rational fraction by -1 is equivalent to changing their signs, and the result is a fraction identically equal to the given one. This transformation has to be used quite often when working with rational fractions.
Thus, if you simultaneously change the signs of the numerator and denominator of a fraction, you will get a fraction equal to the original one. This statement is answered by equality.
Let's give an example. A rational fraction can be replaced by an identically equal fraction with changed signs of the numerator and denominator of the form.
You can do one more thing with fractions: identity transformation, in which the sign of either the numerator or the denominator changes. Let us state the corresponding rule. If you replace the sign of a fraction together with the sign of the numerator or denominator, you get a fraction that is identically equal to the original one. The written statement corresponds to the equalities and .
Proving these equalities is not difficult. The proof is based on the properties of multiplication of numbers. Let's prove the first of them: . Using similar transformations, the equality is proved.
For example, a fraction can be replaced by the expression or.
To conclude this point, we present two more useful equalities and . That is, if you change the sign of only the numerator or only the denominator, the fraction will change its sign. For example, 
And 
.
The considered transformations, which allow changing the sign of the terms of a fraction, are often used when transforming fractional rational expressions.
Reducing rational fractions
The following transformation of rational fractions, called reduction of rational fractions, is based on the same basic property of a fraction. This transformation corresponds to the equality , where a, b and c are some polynomials, and b and c are non-zero.
From the above equality it becomes clear that reducing a rational fraction implies getting rid of common multiplier in its numerator and denominator.
Example.
Cancel a rational fraction.
Solution.
The common factor 2 is immediately visible, let’s perform a reduction by it (when writing, it is convenient to cross out the common factors that are being reduced by). We have 
. Since x 2 =x x and y 7 =y 3 y 4 (see if necessary), it is clear that x is a common factor of the numerator and denominator of the resulting fraction, as is y 3. Let's reduce by these factors: 
. This completes the reduction.
Above we carried out the reduction of rational fractions sequentially. Or it was possible to perform the reduction in one step, immediately reducing the fraction by 2 x y 3. In this case, the solution would look like this: 
.
Answer:
.
When reducing rational fractions, the main problem is that the common factor of the numerator and denominator is not always visible. Moreover, it does not always exist. In order to find a common factor or verify its absence, you need to factor the numerator and denominator of a rational fraction. If there is no common factor, then the original rational fraction does not need to be reduced, otherwise, reduction is carried out.
Various nuances can arise in the process of reducing rational fractions. The main subtleties are discussed in the article reducing algebraic fractions using examples and in detail.
Concluding the conversation about the reduction of rational fractions, we note that this transformation is identical, and the main difficulty in its implementation lies in factoring the polynomials in the numerator and denominator.
Representation of a rational fraction as a sum of fractions
Quite specific, but in some cases very useful, is the transformation of a rational fraction, which consists in its representation as the sum of several fractions, or the sum of an entire expression and a fraction.
A rational fraction, the numerator of which contains a polynomial representing the sum of several monomials, can always be written as a sum of fractions with the same denominators, the numerators of which contain the corresponding monomials. For example, 
. This representation is explained by the rule for adding and subtracting algebraic fractions with like denominators.
In general, any rational fraction can be represented as a sum of fractions in many different ways. For example, the fraction a/b can be represented as the sum of two fractions - an arbitrary fraction c/d and a fraction equal to the difference between the fractions a/b and c/d. This statement is true, since the equality holds 
. For example, a rational fraction can be represented as a sum of fractions different ways: Let's imagine the original fraction as the sum of an integer expression and a fraction. By dividing the numerator by the denominator with a column, we get the equality 
. The value of the expression n 3 +4 for any integer n is an integer. And the value of a fraction is an integer if and only if its denominator is 1, −1, 3, or −3. These values correspond to the values n=3, n=1, n=5 and n=−1, respectively.
Answer:
−1 , 1 , 3 , 5 .
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