Adding and subtracting algebraic fractions with different denominators. Adding and subtracting algebraic fractions with different denominators (8th grade)

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 (to do this, divide the new denominator by the old one).

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

4) Add fractions with like denominators

(to add fractions with the same denominators, you need to add their numerators, but leave the denominator the same).

Examples of adding algebraic fractions.

The lowest common denominator consists of all factors taken to their greatest power. In this case it is equal to ab.

To find an additional factor for each fraction, 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 you need to try them. 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 the factors included in the denominator and is equal to 5x(x-5).

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

(If you don’t like division, you can do it differently. We reason like this: what do you need to multiply the old denominator by to get a new one? To get 5x(x-5) from x(x-5), you need to multiply the first expression by 5. So that 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 complete 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 cannot be factorized, so the common denominator of these fractions is equal to the product of the denominators m(m+3):

Polynomials in the denominators of fractions. In the denominator of the first fraction we take out the common factor x, in the denominator of the second fraction - 2:

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

Algorithm for adding (subtracting) algebraic fractions

1. Reduce all fractions to a common denominator; if they had the same denominators from the very beginning, then this step of the algorithm is omitted.
2. Add (subtract) the resulting fractions with the same denominators.

Example 1. Follow these steps:

A) ; b) ; V) .

Solution. For each pair of algebraic fractions given here, the common denominator was found above, in the lesson “Basic properties of algebraic fractions.” Based on the above example, we get:

The most difficult thing in the above algorithm is, of course, the first step: finding a common denominator and reducing fractions to a common denominator. In Example 1, you may not have felt this difficulty, since we used ready-made results from § 2.
To develop a rule for finding a common denominator, let's analyze example 1.

For fractions and the common denominator is the number 15 - it is divisible by both 3 and 5, and is their common multiple (even the least common multiple).

For fractions, the common denominator is the monomial. It is divided by both and by, i.e., by both monomials, which serve as denominators of fractions. Please note: the number 12 is the least common multiple of the numbers 4 and 6. The variable appears in the denominator of the first fraction with an exponent of 2, in the denominator of the second fraction with an exponent of 3. This largest value of the exponent 3 appears in the common denominator.
For fractions and the common denominator is the product - it is divisible by both the denominator and the denominator.
When finding a common denominator, it is necessary, naturally, to factorize all given denominators (if this was not prepared in the condition). And then you should work in stages: find the least common multiple for numerical coefficients (we are talking about integer coefficients), determine for each several times occurring letter factor the largest exponent, collect all this into one product.
Now you can design the corresponding algorithm.

Algorithm for finding a common denominator for several algebraic fractions

Factor all denominators (numerical coefficients, powers of variables, binomials, trinomials).

Find the least common multiple of the numerical coefficients present in the factorizations compiled in the first step.

Compose a product by including as factors all the letter factors of the expansions obtained in the first step of the algorithm. If a certain factor (power of a variable, binomial, trinomial) is present in several expansions, then it should be taken with an exponent equal to the largest of the available ones.

Add to the product obtained in the third step the numerical coefficient found in the second step; the end result is a common denominator.

Comment. In fact, you can find as many common denominators for two algebraic fractions as you like. For example, for fractions And the common denominator can be the number 30, the number 60, and even a monomial . The fact is that 30, and 60, and can be divided by either 3 or 5. For fractions And common denominator, except for the monomial found above , maybe And . What is the monomial better than , how ? It is simpler (in appearance). It is sometimes called not even the common denominator, but the lowest common denominator. Thus, the given algorithm is an algorithm for finding the simplest common denominator of several algebraic fractions, an algorithm for finding the lowest common denominator.

Let's go back to example 1, a. To add algebraic fractions and , it was necessary not only to find a common denominator (the number 15), but also to find additional factors for each of the fractions that would allow the fractions to be brought to a common denominator. For a fraction, such an additional factor is the number 5 (the numerator and denominator of this fraction are additionally multiplied by 5), for a fraction - the number 3 (the numerator and denominator of this fraction are additionally multiplied by 3). An additional factor is the quotient of dividing the common denominator by the denominator of a given fraction.

Typically the following notation is used:

Let's go back to example 1.6. The common denominator for fractions is the monomial. The additional factor for the first fraction is equal (since ), for the second fraction it is equal to 2 (since ). This means that the solution to Example 1.6 can be written as follows:

An algorithm for finding a common denominator for several algebraic fractions was formulated above. But experience shows that this algorithm is not always clear to students, so we will give a slightly modified formulation.

The rule for reducing algebraic fractions to a common denominator

Factor all denominators.

From the first denominator write out the product of all its factors, from the remaining denominators add the missing factors to this product. The resulting product will be the common (new) denominator.

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.

Find a new numerator for each fraction: this will be the product of the old numerator and an additional factor.

Write each fraction with a new numerator and a new (common) denominator.

Example 2. Simplify an expression .

Solution.
First stage. Let's find the common denominator and additional factors.
We have

We take the first denominator in its entirety, and from the second we add a factor that is not in the first denominator. Let's get a common denominator.

It is convenient to arrange records in the form of a table:

Denominators	Common denominator	Additional multipliers

Second stage.
Let's perform the transformations:

If you have some experience, you can skip the first stage and perform it simultaneously with the second stage.
In conclusion, let's look at a more complex example (for those interested).

Example 3. Simplify an expression

Solution. First stage.
Let's factorize all the denominators:

We take the first denominator in its entirety, from the second we take the missing factors and (or), from the third we take the missing factor (since the third denominator contains the factor ).

Denominators	Common denominator	Additional multipliers

Ordinary fractions.

Adding algebraic fractions

Remember!

You can only add fractions with the same denominators!

You can't add fractions without conversions

You can add fractions

When adding algebraic fractions with like denominators:

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

Let's look at an example of adding algebraic fractions.

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

Let's add the numerator of the first fraction with the numerator of the second fraction, and leave the denominator the same. When adding fractions in the resulting numerator, we present similar ones.

Subtracting algebraic fractions

When subtracting algebraic fractions with like denominators:

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

Important!

Be sure to include the entire numerator of the fraction you are subtracting in parentheses.

Otherwise, you will make a mistake in the signs when opening the brackets of the fraction you are subtracting.

Let's look at an example of subtracting algebraic fractions.

Since both algebraic fractions have a denominator of “2c”, this means that these fractions can be subtracted.

Subtract the numerator of the second fraction “(a − b)” from the numerator of the first fraction “(a + d)”. Don't forget to enclose the numerator of the fraction being subtracted in parentheses. When opening parentheses, we use the rule for opening parentheses.

Reducing algebraic fractions to a common denominator

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

Fractions cannot be added in this form because they have different denominators.

Before adding algebraic fractions, they must be 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 will be divided without a remainder into each of the previous denominators of the fractions.

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

We work with numerical coefficients. We determine the LCM (least common multiple) for all numerical coefficients.
We work with polynomials. We define all the different polynomials in the greatest powers.
The product of the numerical coefficient and all various polynomials in the greatest powers will be the common denominator.
Determine what you need to multiply each algebraic fraction by to get a common denominator.

Let's return to our example.

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

We work with numerical coefficients. Find the LCM (the least common multiple is a number that is divisible by each numerical coefficient without a remainder). For "15" and "3" it is "15".
We work with polynomials. It is necessary to list all polynomials in the greatest powers. In the denominators "15a" and "5" there are only
one monomial - “a”.
Let’s multiply the LCM from step 1 “15” and the monomial “a” from step 2. We get “15a”. This will be the common denominator.
For each fraction, we ask ourselves the question: “What should we multiply the denominator of this fraction by to get “15a”?”

Let's look at the first fraction. This fraction already has a denominator of “15a,” which means it doesn’t need to be multiplied by anything.

Let's look at the second fraction. Let’s ask the question: “What do you need to multiply “3” by to get “15a”?” The answer is “5a”.

When reducing a fraction to a common denominator, multiply by “5a” both numerator and denominator.

A shortened notation for reducing an algebraic fraction to a common denominator can be written using “houses”.

To do this, keep the common denominator in mind. Above each fraction at the top “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.

Let's look at an example of subtracting fractions with different denominators.

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

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

Adding and subtracting algebraic fractions using abbreviated multiplication formulas

In some examples, abbreviated multiplication formulas must be used to reduce algebraic fractions to a common denominator.

Let's look at an example of adding algebraic fractions, where we will 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 is clear 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)”.

ADDING AND SUBTRACTING ALGEBRAIC FRACTIONS WITH DIFFERENT DENOMINATORS

Addition and subtraction of algebraic fractions with different denominators is performed using the same algorithm that is used for adding and subtracting ordinary fractions with different denominators: first, the fractions are brought to a common denominator using the corresponding additional factors.
tel, and then add or subtract the resulting fractions with the same denominators according to the rule from § 3. An algorithm can be formulated that covers any cases of addition (subtraction) of algebraic fractions.

Algorithm for adding (subtracting) algebraic fractions

Example 1. Follow these steps:

Solution. For each pair of algebraic fractions given here, the common denominator was found above, in the example from § 2. Based on the above example, we obtain:

To develop a rule for finding a common denominator, let's analyze example 1.
For fractions, the common denominator is the number 15; it is divisible by both 3 and 5, and is their common multiple (even the least common multiple).
For fractions, the common denominator is the monomial 12b 3. It is divisible by both 4b 2 and 6b 3, i.e., by both monomials that serve as denominators of the fractions.

Please note that the number 12 is the least common multiple of the numbers 4 and 6. The variable b is included in the denominator of the first fraction with exponent 2, in the denominator
the second fraction - with exponent 3. This highest value of exponent 3 appears in the common denominator.
For fractions

the common denominator is the product (x + y)(x - y) - it is divided by both the denominator x + y and the denominator x-y.

When finding a common denominator, it is necessary, naturally, to factorize all given denominators (if this was not prepared in the condition). And then you should work in stages: find the least common multiple for numerical coefficients (we are talking about integer coefficients), determine for each several times occurring letter factor the largest exponent, collect all this into one product.

Now you can design the corresponding algorithm.

Algorithm for finding a common denominator for several algebraic fractions

Before moving on, try applying this algorithm to the common denominator rationale for algebraic fractions in Example 1.
Comment. In fact, you can find as many common denominators for two algebraic fractions as you like. For example, for fractions the common
the denominator can be the number 30, or the number 60, or even the monomial 15a2b. The fact is that 30, 60, and 15a 2 b can be divided by either 3 or 5. For
fractions -
the common denominator, in addition to the monomial 12b found above, can be 24b 3 and 48a 2 b 4. Why is the monomial 12b 3 better than 24b 3, than 48a 2 b 4? It is simpler (in appearance). It is sometimes called not even the common denominator, but the lowest common denominator. Thus, the given algorithm is the algorithm
finding the simplest common denominator of several algebraic fractions, an algorithm for finding the lowest common denominator.

Let's go back to example 1, a. To add algebraic fractions, it was necessary not only to find a common denominator (the number 15), but also to find additional factors for each of the fractions that would allow the fractions to be brought to a common denominator. For a fraction, such an additional multi-
the resident is the number 5 (the numerator and denominator of this fraction are additionally multiplied by 5), for the fraction the number is 3 (the numerator and denominator of this fraction are additionally multiplied by 3).

An additional factor is the quotient of dividing the common denominator by the denominator of a given fraction.
Typically the following notation is used:

Let's go back to example 1.6. The common denominator for fractions is the monomial 12b 3. The additional factor for the first fraction is equal to 3b (since 12b 3: 4b 2 = 3 b), for the second fraction it is equal to 2 (since 12b 3: 6b 3 = 2). This means that the solution to Example 1.6 can be written as follows:

The rule for reducing algebraic fractions to a common denominator

Example 2. Simplify an expression

Solution.
First stage. Let's find the common denominator and additional factors.

We have
4a 2 - 1 = (2a - 1) (2a + 1),
2a 2 + a = a(2a + 1).
We take the first denominator in its entirety, and from the second we add the factor a, which is not in the first denominator. Let's get a common denominator

a(2a - 1) (2a +1).

It is convenient to arrange records in the form of a table:

Second stage.
Let's perform the transformations:

If you have some experience, you can skip the first stage and perform it simultaneously with the second stage.

In conclusion, let's look at a more complex example (for those interested).

Example 3 . Simplify an expression

Solution.
First stage.
Let's factorize all the denominators:

1) 2a 4 + 4a 3 b + 2a 2 b 2 = 2a 2 (a 2 + 2ab + b 2) = 2a 2 (a + b) 2;

2) 3ab 2 - For 3 = For (b 2 - a 2) = For (b - a) (b + a);

3) 6a 4 -6a 3 b = 6a 3 (a-b).

We take the first denominator in its entirety, from the second we take the missing factors 3 and b - a (or a - b), from the third we take the missing factor a (since the third denominator contains the factor a 3).

Algebraic fractions

Note that if an additional factor has a “-” sign, it is usually placed before the entire fraction, i.e., the sign will have to be changed before the second fraction.

Second stage.
Let's perform the transformations:

Note that replacing the expression given in Example 3 with the resulting algebraic fraction is an identical transformation for acceptable values of the variables. In this case, any values of the variables a and b are acceptable, except a = 0, a = b, a = - b (in these
cases, the denominators go to zero).

The video lesson “Addition and subtraction of algebraic fractions with different denominators” is a visual aid that provides theoretical material, explains in detail the algorithms and features of performing operations of subtraction and addition of fractions with different denominators. With the help of the manual, it is easier for the teacher to develop the students' ability to perform operations with algebraic fractions. During the video lesson, a number of examples are considered, the solution of which is described in detail, paying attention to important details.

The use of a video lesson in a mathematics lesson allows the teacher to quickly achieve educational goals and increase the effectiveness of teaching. The clarity of the demonstration helps students remember the material and master it more deeply, so the video can be used to accompany the teacher’s explanation. If this video is used as part of a lesson, then the teacher’s time is freed up to enhance individual work and use other teaching tools to improve teaching efficiency.

The demonstration begins by introducing the topic of the video lesson. It is noted that performing operations of subtraction and addition of algebraic fractions is similar to performing operations with ordinary fractions. The mechanism of subtraction and addition for ordinary fractions is reminiscent - the fractions are brought to a common denominator, and then the operations themselves are performed directly.

The algorithm for subtracting and adding algebraic fractions is voiced and described on the screen. It consists of two steps - reducing fractions to like denominators and then adding (or subtracting) fractions with equal denominators. The application of the algorithm is considered using the example of finding the values of the expressions a/4b 2 -a 2 /6b 3 , as well as x/(x+y)-x/(x-y). It is noted that to solve the first example it is necessary to reduce both fractions to the same denominator. This denominator will be 12b 3. Reducing these fractions to the denominator 12b 3 was discussed in detail in the previous video lesson. As a result of the transformation, two fractions are obtained with equal denominators 3ab/12b 3 and 2a 2 /12b 3. These fractions are added according to the rule for adding fractions with equal denominators. After adding the numerators of the fractions, the result is the fraction (3ab+2a 2)/12b 3. The following describes the solution to the example x/(x+y)-x/(x-y). After reducing the fractions to the same denominator, the resulting fractions are (x 2 -xy)/(x 2 -y 2) and (x 2 +xy)/(x 2 -y 2). According to the rule for subtracting fractions with equal denominators, we perform the operation with the numerators, after which we obtain the fraction -2xy/(x 2 -y 2).

It is noted that the most difficult step in solving problems involving addition and subtraction of fractions with different denominators is bringing them to a common denominator. Tips are given on how to more easily develop skills in solving these problems. The common denominator of a fraction is analyzed. It consists of a numerical coefficient with a variable raised to a power. It can be seen that the expression can be divided into the denominators of the first and second fractions. In this case, the numerical coefficient 12 is the least common multiple of the numerical coefficients of the fractions 4 and 6. And the variable b contains both denominators 4b 2 and 6b 3. In this case, the common denominator contains the variable to the greatest extent among the denominators of the original fractions. Finding the common denominator for x/(x+y) and x/(x-y) is also considered. It is noted that the common denominator (x+y)(x-y) is divided by each denominator. So, solving the problem comes down to finding the least common multiple of the available numerical coefficients, as well as finding the highest exponent for a letter variable that occurs several times. Then, after collecting these parts into a total product, a common denominator is obtained.

An algorithm for finding a common denominator for several fractions is announced and formulated on the screen. This algorithm consists of four stages, in the first of which the denominators are factorized. At the second stage of the algorithm, the least common multiple of the available coefficients included in the denominators of the fractions is found. At the third stage, a product is compiled, which includes letter factors of the denominator decompositions, while the letter exponent present in several denominators is selected to the greatest extent. At the fourth stage, the numerical and letter factors found in the previous stages are collected into one product. This will be the common denominator. A note is made about the considered algorithm. In the example of finding the common denominator of the fractions a/4b 2 and a 2 /6b 3, it is noted that in addition to 12b 3 there are other denominators 24b 3 and 48a 2 b 3. And for every set of fractions you can find many common denominators. However, the denominator 12b 3 is the simplest and most convenient, so it is also called the least common denominator of the original fractions. Additional factors are the result of the partial common denominator and the original denominator of the fraction. It is demonstrated in detail using animation how the numerator and denominator of fractions are multiplied by an additional factor.

Next, it is proposed to consider the algorithm for reducing algebraic fractions to a common denominator in a simpler form, so that it is more understandable for students. It also consists of four steps, the first of which is factorization of the denominators. Then it is proposed to write out all the factors from the first denominator, and supplement the product with the missing factors from the remaining denominators. This way the common denominator is found. Additional factors are found for each fraction from those factors of the denominator that do not fall into the common denominator. The fourth step is to determine for each fraction a new numerator, which is the product of the old numerator and an additional factor. Then each fraction is written with a new numerator and denominator.

The following example describes a simplification of the expression 3a/(4a 2 -1)-(a+1)/(2a 2 +a). At the first stage of solving, the denominators of each fraction are factored. For products, the common factor is (2a+1). By supplementing the product with the remaining factors (2a-1) and a, we obtain a common denominator of the form a(2a-1)(2a+1). An auxiliary table is constructed in which the common denominator, denominators, and additional factors are indicated. At the second stage of the solution, each numerator is multiplied by an additional factor, and subtraction is performed. The result is the fraction (a 2 -a+1)/a(2a-1)(2a+1).

Example 3 considers a simplification of the expression b/(2a 4 +4a 3 b+2a 2 b 2)-1/(3ab 2 -3a 3)+b/(6a 4 -6a 3 b). The solution is also analyzed step by step, attention is drawn to the essential features of performing operations, the reduction of fractions to a common denominator, and the performance of operations with the numerator are described in detail. As a result of calculations and after transformation, the fraction (2a 3 +6a 2 b-ab 2 +b 3)/6a 3 (a-b)(a+b) 2 is obtained.

The video lesson “Adding and subtracting algebraic fractions with different denominators” can serve as a means of increasing the effectiveness of a mathematics lesson on this topic. The manual will be useful to a teacher providing distance learning for a visual presentation of educational material. For students, the video lesson can be recommended for self-study, as it explains in detail and clearly the features of performing the operations being studied.