Addition and subtraction of algebraic fractions with different denominators. Addition and subtraction of algebraic fractions: rules, examples

This lesson will introduce many different examples, so get paper and pen ready to try to solve them yourself, or at least repeat the solution of each example yourself.
We are studying fractional rational expressions and of particular interest to us are rational fractions, that is, fractions whose numerator and denominator are literal expressions.
The topic of the lesson is “the sum and difference of fractions” and first we will talk about fractions with the same denominators.

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When fractions have the same denominators, then when adding (subtracting) them, you need to perform the indicated actions only with numerators, and leave the denominator the same. Let's look at a few examples. (2 examples on the board - immediately). Now pause to stop the lesson and try to complete these tasks on your own.

Let's move on to actions with fractions that have different denominators. And, the simplest case is opposite denominators. For example, is the sum of fractions with opposite denominators and, meeting with such examples, use the rule:

“The minus sign in the numerator or denominator can be written before the fraction; and vice versa: if the minus sign is written before a fraction, then it can be written either in the numerator or in the denominator.

Let's use it: in the denominator of the second fraction, we take out the "minus" bracket, now this "minus" can be put in front of the fraction, and the denominators will become the same.

Think about what was done in solving this example: before performing the action, the rational fractions were changed so that their denominators became the same. Remember: after all, this is what they do with numerical fractions - they are reduced to a common denominator, using the basic property of a fraction for this. The same principle applies when performing operations with any rational fractions.

(The teacher is on the background of a half-length blackboard.) And again, consider a few examples of performing addition and subtraction with fractions. Pause and think about how to deal with these examples on your own, and then we will check. (on the board - only the conditions of the examples)

In the lesson, consider one task with a special wording: "prove that the expressions are identically equal to each other." On the board are the expressions whose equality needs to be proved.

Let's summarize the lesson:

Topic: "Sum and Difference of Fractions". To find both the sum and the difference, you need to convert the fractions so that they have the same denominators. And after that, you need to perform these actions only with numerators, and leave the denominator the same. The result obtained must be reduced.

When performing addition and subtraction of fractions, use the decomposition of a polynomial into factors. For what? 1) To find the simplest common denominator. 2) To reduce fractions.

This lesson is over, but you have to complete a large number of independent exercises in order to firmly master the topic of today's lesson.

This lesson covers addition and subtraction of rational numbers. The topic is classified as complex. Here it is necessary to use the entire arsenal of previously acquired knowledge.

The rules for adding and subtracting integers are also valid for rational numbers. Recall that rational numbers are numbers that can be represented as a fraction, where a - is the numerator of a fraction b is the denominator of the fraction. Wherein, b should not be null.

In this lesson, we will increasingly refer to fractions and mixed numbers as one common phrase - rational numbers.

Lesson navigation:

Example 1 Find the value of an expression:

We enclose each rational number in brackets along with its signs. We take into account that the plus which is given in the expression is the sign of the operation and does not apply to fractions. This fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:

This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract a smaller module from a larger module, and put the sign of the rational number whose module is larger in front of the answer. And in order to understand which module is greater and which is less, you need to be able to compare the modules of these fractions before calculating them:

The modulus of a rational number is greater than the modulus of a rational number. Therefore, we subtracted from . Got an answer. Then, reducing this fraction by 2, we got the final answer.

Some primitive actions, such as putting numbers in brackets and putting down modules, can be skipped. This example can be written in a shorter way:

Example 2 Find the value of an expression:

We enclose each rational number in brackets along with its signs. We take into account that the minus between rational numbers and is the sign of the operation and does not apply to fractions. This fraction has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:

Let's replace subtraction with addition. Recall that for this you need to add to the minuend the number opposite to the subtrahend:

We got the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus before the answer:

Note. It is not necessary to enclose every rational number in parentheses. This is done for convenience, in order to clearly see what signs rational numbers have.

Example 3 Find the value of an expression:

In this expression, the fractions have different denominators. To make it easier for ourselves, let's bring these fractions to a common denominator. We will not go into detail on how to do this. If you experience difficulties, be sure to repeat the lesson.

After reducing the fractions to a common denominator, the expression will take the following form:

This is the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the received answer we put the sign of the rational number, the module of which is greater:

Let's write down the solution of this example in a shorter way:

Example 4 Find the value of an expression

Compute given expression in the following: we add the rational numbers and , then subtract the rational number from the result obtained.

First action:

Second action:

Example 5. Find the value of an expression:

Let's represent the integer −1 as a fraction, and translate the mixed number into an improper fraction:

We enclose each rational number in brackets along with its signs:

We got the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the received answer we put the sign of the rational number, the module of which is greater:

Got an answer.

There is also a second solution. It consists in putting together whole parts separately.

So, back to the original expression:

Enclose each number in parentheses. For this mixed number temporarily:

Let's calculate the integer parts:

(−1) + (+2) = 1

In the main expression, instead of (−1) + (+2), we write the resulting unit:

The resulting expression. To do this, write the unit and the fraction together:

Let's write the solution in this way in a shorter way:

Example 6 Find the value of an expression

Convert the mixed number to an improper fraction. We rewrite the rest without change:

We enclose each rational number in brackets along with its signs:

Let's replace subtraction with addition:

Let's write down the solution of this example in a shorter way:

Example 7 Find value expression

Let's represent the integer −5 as a fraction, and translate the mixed number into an improper fraction:

Let's bring these fractions to a common denominator. After bringing them to a common denominator, they will take the following form:

We enclose each rational number in brackets along with its signs:

Let's replace subtraction with addition:

We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:

Thus, the value of the expression is .

We will decide given example the second way. Let's go back to the original expression:

Let's write the mixed number in expanded form. We rewrite the rest without changes:

We enclose each rational number in brackets together with its signs:

Let's calculate the integer parts:

In the main expression, instead of writing the resulting number −7

The expression is an expanded form of writing a mixed number. Let's write the number −7 and the fraction together, forming the final answer:

Let's write this solution shortly:

Example 8 Find the value of an expression

We enclose each rational number in brackets together with its signs:

Let's replace subtraction with addition:

We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:

Thus, the value of the expression is

This example can be solved in the second way. It consists in adding the whole and fractional parts separately. Let's go back to the original expression:

We enclose each rational number in brackets along with its signs:

Let's replace subtraction with addition:

We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer. But this time we add separately the integer parts (−1 and −2), and the fractional and

Let's write this solution shortly:

Example 9 Find expression expressions

Convert mixed numbers to improper fractions:

We enclose the rational number in brackets together with its sign. A rational number does not need to be enclosed in brackets, since it is already in brackets:

We got the addition of negative rational numbers. We add the modules of these numbers and put a minus before the received answer:

Thus, the value of the expression is

Now let's try to solve the same example in the second way, namely by adding the integer and fractional parts separately.

This time, in order to get a short solution, let's try to skip some actions, such as writing a mixed number in expanded form and replacing subtraction with addition:

Note that the fractional parts have been reduced to a common denominator.

Example 10 Find the value of an expression

Let's replace subtraction with addition:

The resulting expression does not negative numbers which are the main cause of errors. And since there are no negative numbers, we can remove the plus in front of the subtrahend, and also remove the parentheses:

The result is a simple expression that is easy to calculate. Let's calculate it in any way convenient for us:

Example 11. Find the value of an expression

This is the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and put the sign of the rational number, the module of which is greater, in front of the received answers:

Example 12. Find the value of an expression

The expression consists of several rational numbers. According to, first of all, you need to perform the actions in brackets.

First, we calculate the expression , then the expression We add the results obtained.

First action:

Second action:

Third action:

Answer: expression value equals

Example 13 Find the value of an expression

Convert mixed numbers to improper fractions:

We enclose the rational number in brackets along with its sign. A rational number does not need to be enclosed in parentheses, since it is already in parentheses:

Let's give these fractions in a common denominator. After bringing them to a common denominator, they will take the following form:

Let's replace subtraction with addition:

We got the addition of rational numbers with different signs. Let us subtract the smaller module from the larger module, and put the sign of the rational number, the module of which is greater, in front of the received answers:

Thus, the value of the expression equals

Consider the addition and subtraction of decimal fractions, which are also rational numbers and which can be both positive and negative.

Example 14 Find the value of the expression −3.2 + 4.3

We enclose each rational number in brackets along with its signs. We take into account that the plus that is given in the expression is the sign of the operation and does not apply to the decimal fraction 4.3. This decimal has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:

(−3,2) + (+4,3)

This is the addition of rational numbers with different signs. To add rational numbers with different signs, you need to subtract a smaller module from a larger module, and put the rational number whose module is larger in front of the answer. And in order to understand which modulus is larger and which is smaller, you need to be able to compare the moduli of these decimal fractions before calculating them:

(−3,2) + (+4,3) = |+4,3| − |−3,2| = 1,1

The modulus of 4.3 is greater than the modulus of −3.2, so we subtracted 3.2 from 4.3. Got the answer 1.1. The answer is yes, because the answer must be preceded by the sign of the rational number whose modulus is greater. And the modulus of 4.3 is greater than the modulus of −3.2

Thus, the value of the expression −3.2 + (+4.3) is 1.1

−3,2 + (+4,3) = 1,1

Example 15 Find the value of the expression 3.5 + (−8.3)

This is the addition of rational numbers with different signs. As in the previous example, we subtract the smaller one from the larger module and put the sign of the rational number, the module of which is greater, before the answer:

3,5 + (−8,3) = −(|−8,3| − |3,5|) = −(8,3 − 3,5) = −(4,8) = −4,8

Thus, the value of the expression 3.5 + (−8.3) is equal to −4.8

This example can be written shorter:

3,5 + (−8,3) = −4,8

Example 16 Find the value of the expression −7.2 + (−3.11)

This is the addition of negative rational numbers. To add negative rational numbers, you need to add their modules and put a minus before the answer.

You can skip the entry with modules to avoid cluttering up the expression:

−7,2 + (−3,11) = −7,20 + (−3,11) = −(7,20 + 3,11) = −(10,31) = −10,31

Thus, the value of the expression −7.2 + (−3.11) is equal to −10.31

This example can be written shorter:

−7,2 + (−3,11) = −10,31

Example 17. Find the value of the expression −0.48 + (−2.7)

This is the addition of negative rational numbers. We add their modules and put a minus before the received answer. You can skip the entry with modules to avoid cluttering up the expression:

−0,48 + (−2,7) = (−0,48) + (−2,70) = −(0,48 + 2,70) = −(3,18) = −3,18

Example 18. Find the value of the expression −4.9 − 5.9

We enclose each rational number in brackets along with its signs. We take into account that the minus which is located between the rational numbers −4.9 and 5.9 is the sign of the operation and does not apply to the number 5.9. This rational number has its own plus sign, which is invisible due to the fact that it is not written down. But we will write it down for clarity:

(−4,9) − (+5,9)

Let's replace subtraction with addition:

(−4,9) + (−5,9)

We got the addition of negative rational numbers. We add their modules and put a minus before the received answer:

(−4,9) + (−5,9) = −(4,9 + 5,9) = −(10,8) = −10,8

Thus, the value of the expression −4.9 − 5.9 is equal to −10.8

−4,9 − 5,9 = −10,8

Example 19. Find the value of the expression 7 − 9.3

Enclose in brackets each number along with its signs

(+7) − (+9,3)

Let's replace subtraction with addition

(+7) + (−9,3)

(+7) + (−9,3) = −(9,3 − 7) = −(2,3) = −2,3

Thus, the value of the expression 7 − 9.3 is −2.3

Let's write down the solution of this example in a shorter way:

7 − 9,3 = −2,3

Example 20. Find the value of the expression −0.25 − (−1.2)

Let's replace subtraction with addition:

−0,25 + (+1,2)

We got the addition of rational numbers with different signs. We subtract the smaller module from the larger module, and before the answer we put the sign of the number whose module is greater:

−0,25 + (+1,2) = 1,2 − 0,25 = 0,95

Let's write down the solution of this example in a shorter way:

−0,25 − (−1,2) = 0,95

Example 21. Find the value of the expression -3.5 + (4.1 - 7.1)

Perform the actions in brackets, then add the received answer with the number −3.5

First action:

4,1 − 7,1 = (+4,1) − (+7,1) = (+4,1) + (−7,1) = −(7,1 − 4,1) = −(3,0) = −3,0

Second action:

−3,5 + (−3,0) = −(3,5 + 3,0) = −(6,5) = −6,5

Answer: the value of the expression −3.5 + (4.1 − 7.1) is −6.5.

Example 22. Find the value of the expression (3.5 - 2.9) - (3.7 - 9.1)

Let's do the parentheses. Then, from the number that resulted from the execution of the first brackets, subtract the number that resulted from the execution of the second brackets:

First action:

3,5 − 2,9 = (+3,5) − (+2,9) = (+3,5) + (−2,9) = 3,5 − 2,9 = 0,6

Second action:

3,7 − 9,1 = (+3,7) − (+9,1) = (+3,7) + (−9,1) = −(9,1 − 3,7) = −(5,4) = −5,4

Third act

0,6 − (−5,4) = (+0,6) + (+5,4) = 0,6 + 5,4 = 6,0 = 6

Answer: the value of the expression (3.5 - 2.9) - (3.7 - 9.1) is 6.

Example 23. Find the value of an expression −3,8 + 17,15 − 6,2 − 6,15

Enclose in brackets every rational number along with its signs

(−3,8) + (+17,15) − (+6,2) − (+6,15)

Let's replace subtraction with addition where possible:

(−3,8) + (+17,15) + (−6,2) + (−6,15)

The expression consists of several terms. According to the associative law of addition, if the expression consists of several terms, then the sum will not depend on the order of actions. This means that the terms can be added in any order.

We will not reinvent the wheel, but add all the terms from left to right in the order in which they appear:

First action:

(−3,8) + (+17,15) = 17,15 − 3,80 = 13,35

Second action:

13,35 + (−6,2) = 13,35 − −6,20 = 7,15

Third action:

7,15 + (−6,15) = 7,15 − 6,15 = 1,00 = 1

Answer: the value of the expression −3.8 + 17.15 − 6.2 − 6.15 is equal to 1.

Example 24. Find the value of an expression

Let's translate decimal−1.8 to a mixed number. We will rewrite the rest without change:

ALGEBRA
All lessons for grade 8

Lesson #7

Topic. Adding and subtracting fractions with different denominators

Objective: To ensure that students understand the content of the concept of "(lowest) common denominator" for data rational fractions, the content of the algorithm for finding the least common denominator for rational fractions, as well as the algorithm for adding and subtracting rational fractions with different denominators; to form the ability to reproduce the studied algorithms and perform actions with these algorithms to write the sum or difference of rational fractions with different denominators in the form of an (irreducible) rational fraction.

Type of lesson: mastering knowledge, skills and abilities.

Visualization and equipment: reference abstract "Addition and subtraction of rational fractions."

During the classes

I. Organizational stage

II. Examination homework

At the beginning of the lesson, the teacher collects notebooks with completed homework for verification (in order to check students' knowledge and skills on the topic “Addition and subtraction of fractions with the same denominators” and, subject to successful completion, evaluate the work of students) or, organizing the work of students to check home tasks according to the model and adjusting possible mistakes, invites students to complete tasks of a similar content (test work No. 3).

Test work #3

Option 1

1. What is the amount?

3. Find the sum of fractions and.

Option 2

1. What is the sum of fractions?

2. Find the difference between fractions and.

3. Find the sum of fractions and.

4. Find the sum of fractions .

III. Formulation of revenge and lesson objectives

Conscious perception of the purpose of the lesson can be facilitated by a conversation, during which students will answer such questions from the teacher:

1. How to add (subtract) ordinary fractions with the same denominators?

2. How is the addition (subtraction) of fractions with different denominators performed?

3. How to add (subtract) rational fractions with the same denominators? Is this rule similar to the corresponding rule for fractions?

4. Can a rational fraction be represented as an equal rational fraction with a different denominator? How to do this (what is the name of such an action and what is the mechanism for its implementation)?

After the end of the conversation, students should realize that it is important to study the addition and subtraction of rational fractions with different denominators. The study of the issue of the possibility of implementation and the algorithm for adding (subtracting) rational fractions with different denominators with the main didactic purpose of the lesson.

IV. Updating of basic knowledge and skills

@ In accordance with the points discussed at the previous stage, before studying new material, students should activate the knowledge and skills of students in performing addition and subtraction of fractions with different denominators, factoring polynomials, raising a rational fraction to a new denominator, as well as converting the sum or difference of rational fractions to rational fraction.

Doing oral exercises

1. Reduce fractions: ; ; ; ; to the denominator 42.

2. Express expressions as a product:

a) 10x + 15y; b) a2 - 25; c) 42y2 - 21y; d) 7x2 - 7y2; e) 6m - 2n; in) 16x - xy; g) a2 - 4a + 4; c) a8 - a 7.

3. Which denominator is the least common denominator for fractions: a) and; b) and; in and ?

4. What number should be substituted instead of * to form an identity: a) ; b) ; in) ; G) ?

V. Assimilation of knowledge

Plan for learning new material

1. The concept of a common denominator for rational fractions.

2. Algorithms for raising fractions to a common denominator.

3.* General rule addition and subtraction of rational fractions with different denominators.

@ Studying the issue of addition and subtraction of rational fractions with different denominators should begin just with the formation of students' ideas about the content of the concept of the least common denominator of given rational fractions and how to find it. In this case, for clarity, you can use the knowledge of students on how to find the lowest common denominator of fractions and the algorithm for rationally raising a fraction to a new denominator (see above). Having considered typical cases of finding a common denominator for rational fractions, we can generalize the observations by compiling an algorithm for finding the least common denominator for the proposed rational fractions. An algorithm has been compiled that should be “tested” on the previously considered examples. After studying the issue of finding a common denominator, we repeat the algorithm for raising rational fractions to a new denominator and combine them into general image actions called "raising rational fractions to a common denominator."

Having considered the question of raising rational fractions to a common denominator, we proceed to study the question of applying these actions while adding or subtracting rational fractions with different denominators: the algorithm for adding and subtracting rational fractions with different denominators consists. At the same time, it should be emphasized that this algorithm is based on the well-known algorithm for adding and subtracting rational fractions with the same denominators, to which the algorithm for raising rational fractions to a common denominator has been added.

During the study of the topic, difficulties may arise, due, among other things, to the fact that the addition and subtraction of fractions with different denominators involves a longer sequence of actions, which requires sufficiently developed attention of students and the ability to switch from one algorithm to another. At the same time, it should be noted that in some students at the beginning of the study of the topic, difficulties arise precisely because the named psychological mechanisms are not yet sufficiently developed. Therefore, the teacher, already based on the knowledge of the level of preparation of students, can decide whether this lesson should study the algorithms for raising fractions to a common denominator and adding and subtracting fractions with different denominators, focusing on this lesson only on one algorithm for raising fractions to a new denominator and work out stable skills in its application, and in the next lesson, start studying the algorithm for adding and subtracting fractions with different denominators (see 3) of the plan).

VI. Mastering skills

Doing oral exercises

Find the smallest common denominator of the fractions:

a) and; b) and; in and ; d) and; e) and; e) and .

From the named pairs of fractions, choose those that have a common denominator:

a) the product of their denominators;

b) one of the denominators of the two fractions presented;

c) an expression composed of all the different factors of the denominators of these fractions.

Performing written exercises

@ *To realize the didactic goal in this lesson, you should solve the following tasks.

1. Reduction to the (least) common denominator of a rational fraction.

1) Reduce to a common denominator of the fraction:

a) and; b) and; in and ; d) and; e) and; f) and g) and; c) and .

2) Reduce to a common denominator of the fraction:
a) and; b) and; in and ; d) and .

2. Reducing to the (lowest) common denominator and adding or subtracting rational fractions with different denominators.

1) Express as a fraction:

a) ; b) ; in) ; G) ; e) ; e).

2) Perform addition (subtraction) of fractions:

a) ; b) ; in) ; G) ; e) ; e) .

Lesson of multi-level generalizing repetition on the topic:
"Addition and subtraction of rational fractions"

Lesson Objectives:

1. educational - repeat, summarize and systematize the material of the topic. Create conditions for control (self-control) of the assimilation of knowledge and skills.

2. Educational - contribute to the formation of skills to apply techniques: generalizations, highlighting the main thing, transferring knowledge to a life situation; development of mathematical horizons in solving problems, thinking and speech, attention, memory.

3 . Educational - to promote the education of interest in mathematics, activity, general culture.

Lesson type- generalization.

Lesson Form – didactic game"Math Rally"

Methods -Reproductive, partially exploratory

Means of education:

Practical – Computer, screen, textbook, flashcards

intellectual means - analysis, synthesis

emotional means - interest, joy, chagrin.

Activities:

According to the method of implementation - they listened, told, wrote, analyzed, generalized, systematized.

According to the distribution of tasks - frontal, individual, group.

During the classes:

Stages

Time

Goal setting, organizational moment

( Independent work)

Homework assignment. Cards

1 stage of the lesson - organizational (1 minute).

Good afternoon guys! The image of racers on cars and the name "Math Rally" appear on the screen. The topic of the lesson is "Addition and subtraction of rational fractions."

What do you think we will do today? Today we will not have a simple lesson on the topic, but a general one. lesson game"Mathematical Rally". In the lesson we will repeat the addition, subtraction of rational fractions.

The game involves 6 crews. First we need to prepare for the races.

To do this, from each race track, I invite one representative of the crew to the board to choose the car on which you will continue your journey. (Three students solve multi-level speed tasks at the board. Whoever solves it faster gets the fastest car.)

On "3" (Mikhail Kalmykov)

On "4" (Shevchenko Alexandra)

On "5" (Schmalz Alina)

Each crew is given a waybill.

STAGE

Result

Crew preparation for launch (oral work)

Checking the area (Fill in the blanks)

Racing in the city (Math dictation)

Accident, repair (PIT STOP) (Find the mistake)

Rest on a halt. Physical education minute

Cross country racing ( Independent work)

Lesson results. Reflection. Grading

2 stage lesson – CREW PREPARATION FOR LAUNCH "oral work"(5 minutes)Repetition theoretical material on the topic "Decomposition of a polynomial into factors."Teacher: "Let's remember how to factorize polynomials, as we need this to master the main topic of our lesson."Students in an arbitrary sequence name ways to factorize polynomials.Students are then asked to verbally factor out:

Multiply

Answer

Motto

1) 4x + 8

2av (2v + 3a) PI

then

2) 3av - 4as

(5-y)(5+y) IU

ro

3) 4av² + 6a²v

2(x-1)(x+1) LE

pi

4) x² - 9

(y+5) 2 BUT

camping

5) 25 - y²

(x-3) 2 D

me

6) x² - 6x + 9

4(x+2) TO

7) 2х² - 2

4(a-2)(a+2) H

le

8) 4a² - 16

a(3v-4s) RO

9) y² + 10y + 25.

(x-3)(x+3) C

but

ANSWER: "Hurry up slowly!"
Students factorize a polynomial and immediately indicate the decomposition method. 3 stage of the lesson- TERRAIN CHECK (3 minutes).
Exercise: Fill in the blanks

Repetition of theoretical material on the topic "Addition and subtraction of rational fractions."

To add fractions with the same denominators, you need to add them …………………., and ………………… .. leave the same.

A fraction is called rational if ………………… contains……………………..

The value of a fraction will not change if the numerator and denominator of the fraction……………….or ………………….to the same expression………………

To add or subtract fractions with different denominators, you need to …………………. these fractions to the common ………………………

To reduce a rational fraction, you need to………………….and………………..

split into……………………….

To subtract fractions with the same denominators, you need from ……………. subtract the first fraction ………………… the second fraction, and …………………… leave the same

The division of the numerator and denominator by their ….………………………………… is called………………..fractions

The teacher suggests repeating these rules several times, including weakly performing students in the work.Stage 4 lesson – RACING IN THE CITY (Mathematical dictation) -7 minutes

1 crew-verbal (yes, no)

2 crew-digital (yes-1, no-0)

3 crew-graphic (yes_, no ^)

MATHEMATICAL DICTATION:

1. ODZ fractions 5x / (x-3) all numbers except 3

2. The expression 2x-5/12 is a rational fraction

3. This fraction --16/x makes sense for any value of x

4. The smallest common denominator of these fractions7/(x-3) and 15x/(x+3) equals x 2 -9
5. The fraction 5a-10/20a is contractible
6. The numerator and denominator of this fraction 7a-14a 2 /(a 2 -in 2 ) can be decomposed using only FSO
7. The denominator of a fraction cannot be zero
ANSWERS:

In this article, we will analyze in detail addition and subtraction of algebraic fractions. Let's start with adding and subtracting algebraic fractions with the same denominators. After that, we write the corresponding rule for fractions with different denominators. In conclusion, we will show how to add an algebraic fraction to a polynomial and how to perform their subtraction. Traditionally, we will provide all the information with characteristic examples with an explanation of each step of the solution process.

Page navigation.

When the denominators are the same

The principles carry over to algebraic fractions. We know that when adding and subtracting ordinary fractions with the same denominators, their numerators are added or subtracted, and the denominator remains the same. For example, and .

Similarly, it is formulated rule for adding and subtracting algebraic fractions with the same denominators: to add or subtract algebraic fractions with the same denominators, you need to add or subtract the numerators of the fractions, respectively, and leave the denominator unchanged.

It follows from this rule that as a result of adding or subtracting algebraic fractions, a new algebraic fraction is obtained (in a particular case, a polynomial, a monomial, or a number).

Let us give an example of the application of the sounded rule.

Example.

Find the sum of algebraic fractions and .

Solution.

We need to add algebraic fractions with the same denominators. The rule tells us that we need to add the numerators of these fractions, and leave the denominator the same. So, add the polynomials in the numerators: x 2 +2 x y−5+3−x y= x 2 +(2 x y−x y)−5+3=x 2 +x y−2. Therefore, the sum of the original fractions is .

In practice, the solution is usually written briefly in the form of a chain of equalities, reflecting all the actions performed. In our case, the summary of the solution is as follows:

Answer:

.

Note that if, as a result of addition or subtraction of algebraic fractions, a reducible fraction is obtained, then it is desirable to reduce it.

Example.

Subtract a fraction from an algebraic fraction.

Solution.

Since the denominators of algebraic fractions are equal, it is necessary to subtract the numerator of the second from the numerator of the first fraction, and leave the denominator the same: .

It is easy to see that it is possible to perform the reduction of an algebraic fraction. To do this, we transform its denominator by applying difference of squares formula. We have .

Answer:

.

Absolutely similarly add or subtract three and large quantity algebraic fractions with the same denominators. For example, .

Addition and subtraction of algebraic fractions with different denominators

Recall how we perform the addition and subtraction of ordinary fractions with different denominators: first we bring them to a common denominator, and then add these fractions with the same denominators. For example, or .

There is a similar rule for adding and subtracting algebraic fractions with different denominators:

• first, all fractions are reduced to a common denominator;
• after which the addition and subtraction of the resulting fractions with the same denominators is performed.

For the successful application of the voiced rule, you need to understand well the reduction of algebraic fractions to a common denominator. This is what we'll do.

Bringing algebraic fractions to a common denominator.

Reducing algebraic fractions to a common denominator is an identical transformation of the original fractions, after which the denominators of all fractions become the same. It is convenient to use the following algorithm for reducing algebraic fractions to a common denominator:

• first, a common denominator of algebraic fractions is found;
• further, additional factors are determined for each of the fractions, for which the common denominator is divided by the denominators of the original fractions;
• finally, the numerators and denominators of the original algebraic fractions are multiplied by the corresponding additional factors.

Example.

Give algebraic fractions and to a common denominator.

Solution.

First, let's determine the common denominator of algebraic fractions. To do this, we decompose the denominators of all fractions into factors: 2 a 3 −4 a 2 =2 a 2 (a−2), 3 a 2 −6 a=3 a (a−2) and 4 a 5 −16 a 3 =4 a 3 (a−2) (a+2). From here we find the common denominator 12·a 3 ·(a−2)·(a+2) .

Now we proceed to finding additional factors. To do this, we divide the common denominator by the denominator of the first fraction (it is convenient to take its expansion), we have 12 a 3 (a−2) (a+2):(2 a 2 (a−2))=6 a (a+2). Thus, the additional factor for the first fraction is 6·a·(a+2) . Similarly, we find additional factors for the second and third fractions: 12 a 3 (a−2) (a+2):(3 a (a−2))=4 a 2 (a+2) and 12 a 3 (a−2) (a+2):(4 a 3 (a−2) (a+2))=3.

It remains to multiply the numerators and denominators of the original fractions by the corresponding additional factors:

This completes the reduction of the original algebraic fractions to a common denominator. If necessary, the resulting fractions can be converted to the form of algebraic fractions by multiplying polynomials and monomials in numerators and denominators.

So, we figured out the reduction of algebraic fractions to a common denominator. We are now prepared to perform addition and subtraction of algebraic fractions with different denominators. Yes, we almost forgot to warn you: it is convenient to leave the common denominator in the form of a product until the very last moment - you may have to reduce the fraction that will be obtained after addition or subtraction.

Example.

Perform addition of algebraic fractions and .

Solution.

Obviously, the original fractions have different denominators, so in order to add them, you first need to bring them to a common denominator. To do this, we factor out the denominators: x 2 + x \u003d x (x + 1) , and x 2 +3 x + 2 \u003d (x + 1) (x + 2) , since the roots of the square trinomial x 2 + 3 x+2 are the numbers −1 and −2 . From here we find the common denominator, it has the form x·(x+1)·(x+2) . Then the additional factor of the first fraction will be x + 2, and the second fraction - x.

So, and .

It remains to add the fractions reduced to a common denominator:

The resulting fraction can be reduced. Indeed, if the numerator takes the two out of brackets, then the common factor x + 1 becomes visible, by which the fraction is reduced:.

Finally, we represent the resulting fraction as an algebraic one, for which we replace the product in the denominator with a polynomial: .

Let's make a short solution that takes into account all our reasoning:

Answer:

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And one more thing: it is advisable to pre-transform algebraic fractions before adding or subtracting them in order to simplify them (if, of course, there is such a possibility).

Example.

Subtract algebraic fractions and .

Solution.

Let's perform some transformations of algebraic fractions, perhaps they will simplify the solution process. To begin with, we take out the numerical coefficients of the variables in the denominator: and . It is already interesting - the common factor of the denominators of fractions has become visible.