Addition and subtraction of algebraic fractions: rules, examples. Addition and subtraction of algebraic fractions

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. We will provide all the information according to tradition typical examples explaining each step of the solution process.

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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.

Bringing algebraic fractions to a common denominator is identity transformation initial 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:

.

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.

Ordinary fractions.

Addition of algebraic fractions

Remember!

You can only add fractions with the same denominators!

You can't add fractions without transformations

Can add fractions

When adding algebraic fractions with the same denominators:

1. the numerator of the first fraction is added to the numerator of the second fraction;
2. the denominator remains the same.

Consider an example of adding algebraic fractions.

Since the denominator of both fractions is “2a”, it means that the fractions can be added.

Add the numerator of the first fraction to the numerator of the second fraction, and leave the denominator the same. When adding fractions in the resulting numerator, we present similar ones.

Subtraction of algebraic fractions

When subtracting algebraic fractions with the same denominators:

1. the numerator of the second fraction is subtracted from the numerator of the first fraction.
2. the denominator remains the same.

Important!

Be sure to enclose the entire numerator of the subtracted fraction in brackets.

Otherwise, you will make a mistake in the signs when opening the brackets of the fraction to be subtracted.

Consider an example of subtracting algebraic fractions.

Since both algebraic fractions have a denominator " 2c", It means that these fractions can be subtracted.

Subtract from the numerator of the first fraction "(a + d)" the numerator of the second fraction "(a − b)". Do not forget to enclose the numerator of the subtracted fraction in brackets. When opening brackets, we use the rule of opening brackets.

Reduction of algebraic fractions to a common denominator

Let's consider another example. You need to add algebraic fractions.

You can't add fractions this way because they have different denominators.

Before adding algebraic fractions, they must bring to a common denominator.

The rules for reducing algebraic fractions to a common denominator are very similar to the rules for reducing ordinary fractions to a common denominator. .

As a result, we should get a polynomial that divides without a trace into each former denominator of the fractions.

To reduce algebraic fractions to a common denominator you need to do the following.

1. We work with numerical coefficients. We determine the LCM (least common multiple) for all numerical coefficients.
2. We work with polynomials. We define all different polynomials in the largest powers.
3. The product of the numerical coefficient and all the various polynomials to the highest degrees will be the common denominator.
4. Determine what each algebraic fraction needs to be multiplied by to get a common denominator.

Let's go back to our example.

Consider the denominators "15a" and "3" of both fractions and find a common denominator for them.

1. We work with numerical coefficients. We find the LCM (the least common multiple is the number that is divisible by each numerical coefficient without a remainder). For "15" and "3" - this is "15".
2. We work with polynomials. It is necessary to list all polynomials in the largest powers. The denominators "15a" and "5" have only
   one monomial - "a".
3. We multiply the LCM from item 1 "15" and the monomial "a" from item 2. We will get " 15a". This will be the common denominator.
4. For each fraction, let's ask ourselves the question: "What do you need to multiply the denominator of this fraction to get" 15a "?".

Let's look at the first fraction. In this fraction, the denominator is already “ 15a”, which means that it does not need to be multiplied by anything.

Consider the second fraction. Let's ask the question: "What do you need to multiply" 3"To get" 15a"?" The answer is "5a".

When reducing the fraction to a common denominator, we multiply by " 5a" both numerator and denominator.

An abbreviated notation of bringing an algebraic fraction to a common denominator can be written through "houses".

To do this, keep the common denominator in mind. Above each fraction from above “in the house” we write what we multiply each of the fractions by.

Now that the fractions have the same denominators, the fractions can be added.

Consider an example of subtracting fractions with different denominators.

Consider the denominators "(x − y)" and "(x + y)" of both fractions and find a common denominator for them.

We have two different polynomials in the denominators "(x − y)" and "(x + y)". Their product will be a common denominator, i.e. "(x − y)(x + y)" is the common denominator.

Adding and subtracting algebraic fractions using reduced multiplication formulas

In some examples, to reduce algebraic fractions to a common denominator, one must use the reduced multiplication formulas.

Consider an example of adding algebraic fractions, where we need to use the difference of squares formula.

In the first algebraic fraction, the denominator is "(p 2 − 36)". Obviously, the difference of squares formula can be applied to it.

After decomposing the polynomial "(p 2 − 36)" into the product of polynomials
“(p + 6)(p − 6)”, it can be seen that the polynomial “(p + 6)” is repeated in fractions. This means that the common denominator of the fractions will be the product of polynomials "(p + 6)(p − 6)".

to form the ability to perform actions (addition and subtraction) with algebraic fractions with different denominators, based on the rule of addition and subtraction of ordinary fractions with different denominators;

Equipment: Demonstration material.

Tasks for updating knowledge:

1) +; 2) -;

3) + ; 4) +; 5) -.

1) Algorithm for adding and subtracting ordinary fractions with different denominators.

To add or subtract common fractions with different denominators:

1. Convert these fractions to the smallest common denominator.
2. Add or subtract the resulting fractions.

2) Algorithm for reducing algebraic fractions to a common denominator.

1. Let's find additional factors for each of the fractions: these will be the products of those factors that are in the common (new) denominator, but which are not in the old denominator.

3) Standards for independent work with self-test:

3) Card for the reflection stage.

1. This topic is clear to me.
2. I know how to find additional factors for each of the fractions.
3. I can find new numerators for each of the fractions.
4. In independent work, I succeeded.
5. I was able to understand the reason for the mistake that I made in my independent work.
6. I am satisfied with my work in the classroom.

DURING THE CLASSES

1. Self-determination to activity.

Stage goals:

1. Inclusion of students in learning activities: continuation of the journey through the country "Algebraic Expressions".
2. Determining the content of the lesson: continuing to work with algebraic fractions.

Organization educational process at step 1:

Good morning, guys! We continue our fascinating journey through the country "Algebraic Expressions".

What “inhabitants” of the country did we meet in previous lessons? (With algebraic expressions.)

What can we do with familiar algebraic expressions? (Addition and subtraction.)

Which salient feature algebraic fractions that we already know how to add and subtract? (We add and subtract fractions that have the same denominator.)

Right. But we all understand well together that the skills to perform actions with algebraic fractions that have the same denominators are not enough. What else do you think we need to learn to do? (Perform actions with fractions that have different denominators.)

Well done! Shall we continue our journey then? (Yes!)

2. Actualization of knowledge and fixation of difficulties in activities.

Stage goals:

1. Update knowledge about performing actions with fractions with the same denominators, methods of oral calculations.
2. Fix difficulty.

Organization of the educational process at stage 2:

There are several examples on the board for performing actions with fractions:

5) -=-==.

Students are encouraged to voice their solutions in a loud speech.

In the first example, the guys easily give the correct answer, remembering the algorithm for performing actions with algebraic fractions that have the same denominators.

When the comment on example #2 has already been made, the teacher focuses on example #2:

Guys, look what we have interesting in example number 2? (We not only performed actions with algebraic fractions that have the same denominators, but also performed the reduction of the resulting algebraic fraction: we took the minus sign out of brackets, we got the same factors in the numerator and denominator, by which we subsequently reduced the result.)

It is very good that you have not forgotten that the basic property of a fraction is applicable not only to ordinary, but also to algebraic fractions!

Who will comment on the solution of the following three examples for everyone?

Most likely, there will be a student who can easily solve example number 3.

What did you use when solving example number 3? (The algorithm for adding and subtracting ordinary fractions with different denominators helped me.)

How exactly did you act? (I reduced algebraic fractions to the lowest common denominator of 15 and then added them.)

Wonderful! And how are we doing with the last two examples?

When it comes to the next two examples, the guys (each for themselves) fix the difficulty that has arisen.

The words of the students are something like this:

I find it difficult to complete examples 4-5, because before me are algebraic fractions, not with “same” denominators, and these different denominators include variables (No. 4), and in No. 5 there are literal expressions in the denominators! ..”

Answers to tasks 4–5 were not received.

3. Identification of the place and causes of difficulties and setting the goal of the activity.

Stage goals:

1. Fix distinguishing feature tasks that caused difficulty in learning activities.
2. State the purpose and topic of the lesson.

Organization of the educational process at stage 3:

Guys? Where did the difficulty arise? (In examples 4-5.)

Why, when solving them, are you not ready to discuss the solution and give an answer? (Because the algebraic fractions proposed in these tasks have different denominators, and we are familiar with the algorithm for performing operations with algebraic fractions that have the same denominators.

What else do we need to be able to do? (You need to learn how to add and subtract fractions with different denominators.)

I agree with you. How can we formulate the topic of our today's lesson? (Addition and subtraction of algebraic fractions with different denominators.)

The topic of the lesson is written in notebooks.

4. Building a project to get out of the difficulty.

Purpose of the stage:

1. Children building a new way of doing things.
2. Fixing the algorithm for reducing algebraic fractions to a common denominator.

Organization of the educational process at stage 4:

What is the purpose of our lesson today? (Learn to add and subtract algebraic fractions with different denominators.)

How to be? (For this we have to build an algorithm further work with algebraic fractions.)

What do we need to come up with to achieve the goal of the lesson? (An algorithm for reducing algebraic fractions to a common denominator, so that later we can work according to the usual rule for adding and subtracting fractions with the same denominators.)

The work can be organized in groups, each group is given a piece of paper and a marker. Students can offer their own variants of the algorithm in the form of a list of steps. You have 5 minutes to work. Groups post their options for an algorithm or rule, and then each option is analyzed.

Most likely, one of the students will definitely draw an analogy of their algorithm with the algorithm for adding and subtracting ordinary fractions with different denominators: first, they bring the fractions to a common denominator using the appropriate additional factors, and then add and subtract the resulting fractions with the same denominators.

Subsequently, a single variant is derived from this. It can be like this:

1. We decompose all the denominators into factors.
2. From the first denominator we write out the product of all its factors, from the remaining denominators we assign the missing factors to this product. The resulting product will be the common (new) denominator.
3. Let's find additional factors for each of the fractions: these will be the products of those factors that are in the new denominator, but which are not in the old denominator.
4. Let's find a new numerator for each fraction: it will be the product of the old numerator and an additional factor.
5. Let's write each fraction with a new numerator and a common (new) denominator.

Well, let's apply our rule to complete the unsolved proposed tasks. Each task (4, 5) is spoken in turn by some students of the class, the teacher fixes the solution on the board.

We are simply geniuses! We have built an algorithm for adding and subtracting algebraic fractions with different denominators. By joint efforts, we have eliminated the difficulty, since we now have a real “guide” (algorithm) in the country unknown to us “Algebraic fractions”!

5. Primary consolidation in external speech.

Purpose of the stage:

1. Train the ability to bring algebraic fractions to a common denominator.
2. Organize the pronunciation of the studied content of the rule-algorithm in external speech.

Organization of the educational process at stage 5:

Guys, but we all know well that just looking and knowing the “map of the area” is not a journey. What should we do to penetrate deeper and more into the world of algebraic fractions? (We have to solve examples, and generally practice solving examples, in order to consolidate our new algorithm.)

Quite right. Therefore, I propose to begin our study.

The student verbally pronounces the plan of his decision, the teacher corrects if some inaccuracies are made.

Approximately it sounds like this:

We must choose a number that will be divided by 2 and 5 at the same time. This is the number 10. Then we select the variables to the degree we need. So our new denominator will be 10xy. We select additional multipliers. To the first fraction: 5y, to the second: 2x. We multiply the selected additional factors by each old numerator. We get algebraic fractions with the same denominators, perform the subtraction according to the rule already familiar to us.

I'm happy. And now our big team will split into pairs, and we will continue our interesting path.

No. 133 (a, d). The students work in pairs, saying the solution to each other:

a) +=+= =;

d) +=+= =.

6. Independent work with self-test.

Stage goals:

1. Spend independent work.
2. Perform a self-test against the prepared self-test standard.
3. Students will record difficulties, identify the causes of errors and correct errors.

Organization of the educational process at stage 6:

I carefully observed your work and came to the conclusion that each of you is already ready to independently think about ways and find solutions to examples on our today's topic. Therefore, I offer you a small independent work, after which you will be offered a standard with the correct solution and answer.

No. 134 (a, b): perform work on options.

After the work is completed, a standard check is carried out. When checking solutions, students mark “+” the correct solution, “?” not the right decision. It is desirable that students who make mistakes explain the reason why they did the task incorrectly.

Errors are analyzed and corrected.

So, what difficulties did you meet on your way? (I made a mistake when opening brackets that are preceded by a minus sign.)

What is the reason for this? (Simply due to inattention, but in the future I will be more careful!)

What else seemed difficult? (Did I have a hard time finding additional factors for fractions?)

You should definitely study step 3 of the algorithm in more detail so that such a problem does not arise in the future!

Were there any other difficulties? (And I just did not bring similar terms).

And we'll fix it. When you have done everything that is possible according to the new algorithm, you need to remember the material studied for a long time. In particular, the reduction of similar terms, or the reduction of fractions, etc.

7. Inclusion of new knowledge in the knowledge system.

The purpose of the stage: to repeat and consolidate the algorithm for adding and subtracting algebraic fractions with different denominators studied in the lesson.

8. Lesson reflection.

The purpose of the stage: to fix the new content, evaluate their own activities.

Organization of the educational process at stage 8:

What was our goal at the beginning of the lesson? (Learn how to add and subtract fractions with different denominators.)

What did we come up with to achieve the goal? (An algorithm for adding and subtracting algebraic fractions with different denominators.)

What else did we use? (We factored the denominators, selected LCMs for the coefficients, and additional factors for the numerators.)

Now take some colored pen or felt-tip pen and mark with a “+” sign those statements with the truth of which you agree:

Each student has a card with phrases. Children mark and show to the teacher.

Well done!

Homework: paragraph 4 (textbook); No. 126, 127 (task book).

Lesson topic: Addition and subtraction of algebraic fractions.

Lesson Objectives:

Tutorials:

1. repeat the rules of addition and subtraction numeric fractions with the same denominators
2. introduce rules for adding and subtracting algebraic fractions with the same denominators;
3. to form the ability to perform addition and subtraction with algebraic fractions.

Developing:

1. develop thinking, attention, memory, the ability to analyze, compare, compare;
2. expanding the horizons of students;

1. vocabulary replenishment;

Educational:

1. develop an interest in the subject.
2. Cultivate a culture of intellectual work

Equipment:

1. cards - test tasks;
2. a computer;
3. projector;
4. screen;
5. lesson presentation

Motto:

You can't learn math by watching your neighbor do it!

Slide 2.

Lesson plan.

1. Reporting the purpose and topic of the lesson (2 min);
2. Updating the basic knowledge and skills of students (4 min);
3. Oral work (5 min);
4. Learning new material (8 min);
5. Physical education (2 min);
6. Consolidation of new material (10 min);
7. Multiple choice test (10 min);
8. The result of the lesson, conclusions (2 min);
9. Homework. (2 minutes).

Slide 3.

During the classes.

I. Organizational moment:

1) message of the topic of the lesson;

2) communication of the goals and objectives of the lesson.

II. Knowledge update:

What is an algebraic fraction? Give examples.

What does it mean to reduce an algebraic fraction?

How to bring algebraic fractions to a common denominator?

slide 4.

III. Oral work:

1. Read fractions:
2. Find an expression that is redundant a) (a + c) 2; b) ; in) ; G) .
3. Restore partially erased records: to reduce to a common denominator

Slide 5.

1. find the mistake

slide 6.

1. For each fraction, find the fraction equal to it, using the correspondence number - letter:

1) ; 2) 3) .

A) b); in) .

slide 7.8

IV. Learning new material.
1) Repeat the rules for adding and subtracting numerical fractions with the same denominators. Then verbally solve the following examples:

2) Remember the rules for adding and subtracting polynomials and write the following exercises on the board:

3) Students should suggest rules for doing the following examples written on the board:

The solution of the examples is discussed. If the students cannot cope on their own, the teacher explains.

slide 9.

The rules for adding and subtracting algebraic fractions with the same denominators are written in a notebook.
, .

slide 10.

V. Physical education for the eyes

Exercise 1. Make 15 oscillatory movements of the eyes horizontally from right to left, then from left to right.

Exercise 2. Make 15 oscillatory eye movements vertically up - down and down - up.

Exercise 3. Also 15, but circular rotational movements of the eyes from left to right.

Exercise 4. The same, but from right to left.

Exercise 5. Make 15 circular rotational movements with your eyes, first to the right, then to the left, as if drawing a figure eight laid on its side with your eyes.

VI. Consolidation of new material.
1) Front work.

1) Solve tasks

№ 462 (1,3)

2) Add fractions:

3) Subtract fractions:

4) Perform actions.

Slide 11.

2) Individual work.
Four students perform independent work on the board, proposed on the cards.

Card 1.

Card 2.

Card 3.

Card 4.

The rest in notebooks: Perform addition and subtraction of fractions:
a) b)
in)

VII. Performing work in groups and analyzing the results.

Each group is given test tasks, after completing which they receive a word - the name of a famous mathematician.

Answer table:

Check the quality of the job.

Did you get the name of a famous mathematician from the received letters?

If you answered all the questions correctly, you got an “EXCELLENT” rating!!!

If you made a mistake in one step - not bad, but the scientist would probably be offended. You have been rated "GOOD"!

If you made a mistake in two steps, then you did not listen well to the teacher in the lesson and you will have to read the topic in the algebra textbook. You have been rated "SATISFACTORY".

If you made a mistake in more than two steps, then you did not listen to the teacher at all in the lesson and you will have to read the algebra textbook very carefully. You have been rated "UNSATISFACTORY".

Slide 13-17.

When time is available, tasks are solved:
1. Prove that the expression
for all values ​​of a2 takes positive values.
2. Present a fraction as a sum or difference of an integer expression and a fraction:
a); b) c)

3. Knowing that, find the value of the fraction:
a); b) c)

VIII. Summarizing.

I X. Homework:Read the textbook material p.26, learn the rules of this paragraph. Solve problems No. 462(2,4); make 5 examples for adding and subtracting algebraic fractions; find information about the mathematicians whose names we heard today.

How to perform addition of algebraic (rational) fractions?

To add algebraic fractions, you need:

1) Find the smallest of these fractions.

2) Find an additional factor for each fraction (for this you need to divide the new denominator by the old one).

3) Multiply the additional factor by the numerator and denominator.

4) Perform the addition of fractions with the same denominators

(To add fractions with the same denominators, you must add their numerators, and leave the denominator the same).

Examples of addition of algebraic fractions.

The lowest common denominator is the sum of all factors taken to the highest power. In this case, it is equal to ab.

To find an additional factor to each fraction, we divide the new denominator by the old one. ab:a=b, ab:(ab)=1.

The numerator has a common factor a. We take it out of the bracket and reduce the fraction by a:

The denominators of these fractions are polynomials, so they need to be tried. In the denominator of the first fraction there is a common factor x, in the second - 5. We take them out of brackets:

The common denominator consists of all factors included in the denominator and is equal to 5x(x-5).

To find an additional factor to each fraction, we divide the new denominator by the old one.

(If you don’t like the division, you can do it differently. We argue like this: what do you need to multiply the old denominator to get a new one? To get 5x(x-5) from x (x-5), you need to multiply the first expression by 5. To get from 5 (x-5) to get 5x(x-5), you need to multiply the 1st expression by x. Thus, the additional factor to the first fraction is 5, to the second - x).

The numerator is the full square of the difference. We collapse it according to the formula and reduce the fraction by (x-5):

The denominator of the first fraction is a polynomial. It does not factor into factors, so the common denominator of these fractions is equal to the product of the denominators m (m + 3):

Polynomials in the denominators of fractions,. We take out the common factor x in the denominator of the first fraction, and 2 in the denominator of the second fraction:

The denominator of the first fraction in brackets is the difference of squares.