Multiplying algebraic fractions with different denominators. Multiplying algebraic fractions. How to correctly divide and multiply algebraic fractions

Video lesson “Multiplication and division of algebraic fractions. Raising an algebraic fraction to a power" is an auxiliary tool for teaching a mathematics lesson on this topic. With the help of a video lesson, it is easier for a teacher to develop in students the ability to multiply and divide algebraic fractions. The visual aid contains a detailed, understandable description of examples in which multiplication and division operations are performed. The material can be demonstrated during the teacher's explanation or become a separate part of the lesson.

To develop the ability to solve problems on multiplication and division of algebraic fractions, important comments are given as the solution is described; points that require memorization and deep understanding are highlighted using color, bold font, and pointers. With the help of a video lesson, the teacher can increase the effectiveness of the lesson. This visual aid will help you quickly and effectively achieve your learning goals.

The video lesson begins by introducing the topic. After this, it is indicated that multiplication and division operations with algebraic fractions are performed similarly to operations with ordinary fractions. The screen demonstrates the rules for multiplying, dividing and exponentiating fractions. Multiplication of fractions is demonstrated using letter options. It is noted that when multiplying fractions, the numerators, as well as the denominators, are multiplied. This gives the resulting fraction a/b·c/d=ac/bd. The division of fractions is demonstrated using the expression a/b:c/d as an example. It is indicated that to perform the division operation it is necessary to write in the numerator the product of the numerator of the dividend and the denominator of the divisor. The denominator of a quotient is the product of the denominator of the dividend and the numerator of the divisor. Thus, the division operation turns into an operation of multiplying the fraction of the dividend and the reciprocal of the divisor. Raising a fraction to a power is equivalent to a fraction in which the numerator and denominator are raised to the assigned power.

The solution to the examples is discussed below. In example 1, it is necessary to perform the actions (5x-5y)/(x-y)·(x 2 -y 2)/10x. To solve this example, the numerator of the second fraction included in the product is factorized. Using abbreviated multiplication formulas, the transformation x 2 -y 2 = (x+y)(x-y) is made. Then the numerators of the fractions and denominators are multiplied. After carrying out the operations, it is clear that the numerator and denominator have factors that can be reduced using the basic property of a fraction. As a result of the transformations, the fraction (x+y) 2 /2x is obtained. Here we also consider the execution of actions 7a 3 b 5 /(3a-3b)·(6b 2 -12ab+6a 2)/49a 4 b 5. All numerators and denominators are considered for the possibility of factorization and identification of common factors. Then the numerators and denominators are multiplied. After multiplication, reductions are made. The result of the transformation is the fraction 2(a-b)/7a.

An example is considered in which it is necessary to perform the actions (x 3 -1)/8y:(x 2 +x+1)/16y 2. To solve the expression, it is proposed to transform the numerator of the first fraction using the abbreviated multiplication formula x 3 -1=(x-1)(x 2 +x+1). According to the rule for dividing fractions, the first fraction is multiplied by the reciprocal of the second fraction. After multiplying the numerators and denominators, a fraction is obtained that contains the same factors in the numerator and denominator. They are shrinking. The result is the fraction (x-1)2y. The solution to the example (a 4 -b 4)/(ab+2b-3a-6):(b-a)(a+2) is also described here. Similar to the previous example, the abbreviated multiplication formula is used to convert the numerator. The denominator of the fraction is also converted. The first fraction is then multiplied with the reciprocal of the second fraction. After multiplication, transformations are performed, reducing the numerator and denominator by common factors. The result is the fraction -(a+b)(a 2 +b 2)/(b-3). Students' attention is drawn to how the signs of the numerator and denominator change when multiplying.

In the third example, you need to perform operations with fractions ((x+2)/(3x 2 -6x)) 3:((x 2 +4x+4)/(x 2 -4x+4)) 2 . In solving this example, the rule for raising a fraction to a power is applied. Both the first and second fractions are raised to a power. They are converted by raising the numerator and denominator of the fraction to a power. In addition, to convert the denominators of fractions, the abbreviated multiplication formula is used, highlighting the common factor. To divide the first fraction by the second, you need to multiply the first fraction by the reciprocal of the second. The numerator and denominator form expressions that can be abbreviated. After the transformation, the fraction (x-2)/27x 3 (x+2) is obtained.

Video lesson “Multiplication and division of algebraic fractions. Raising an algebraic fraction to a power" is used to increase the effectiveness of a traditional mathematics lesson. The material may be useful to a teacher teaching remotely. A detailed, clear description of the solutions to the examples will help students who are independently mastering the subject or require additional training.

In this article, we continue to explore the basic operations that can be done with algebraic fractions. Here we will look at multiplication and division: first we will derive the necessary rules, and then we will illustrate them with solutions to problems.

How to correctly divide and multiply algebraic fractions

To multiply algebraic fractions or divide one fraction by another, we need to use the same rules as for ordinary fractions. Let's remember their wording.

When we need to multiply one ordinary fraction by another, we perform separate multiplication of numerators and separate denominators, after which we write down the final fraction, placing the corresponding products in place. An example of such a calculation:

2 3 4 7 = 2 4 3 7 = 8 21

And when we need to divide ordinary fractions, we do this by multiplying by the inverse fraction of the divisor, for example:

2 3: 7 11 = 2 3 11 7 = 22 7 = 1 1 21

Multiplying and dividing algebraic fractions follows the same principles. Let's formulate a rule:

Definition 1

To multiply two or more algebraic fractions, you need to multiply the numerators and denominators separately. The result will be a fraction, the numerator of which will be the product of the numerators, and the denominator will be the product of the denominators.

In literal form, the rule can be written as a b · c d = a · c b · d. Here a, b, c and d will represent certain polynomials, and b and d cannot be zero.

Definition 2

In order to divide one algebraic fraction by another, you need to multiply the first fraction by the reciprocal of the second.

This rule can also be written as a b: c d = a b · d c = a · d b · c. Letters a, b, c and d here means polynomials, of which a, b, c and d cannot be zero.

Let us separately dwell on what a reverse algebraic fraction is. It is a fraction that, when multiplied by the original one, results in one. That is, such fractions will be similar to reciprocal numbers. Otherwise, we can say that the reciprocal algebraic fraction consists of the same values ​​as the original one, but its numerator and denominator change places. So, in relation to the fraction a · b + 1 a 3, the fraction a 3 a · b + 1 will be the inverse.

Solving problems involving multiplication and division of algebraic fractions

In this paragraph we will look at how to correctly apply the above rules in practice. Let's start with a simple and clear example.

Example 1

Condition: multiply the fraction 1 x + y by 3 · x · y x 2 + 5, and then divide one fraction by the other.

Solution

Let's do the multiplication first. According to the rule, you need to multiply the numerators and denominators separately:

1 x + y 3 x y x 2 + 5 = 1 3 x y (x + y) (x 2 + 5)

We have received a new polynomial that needs to be brought to standard form. Let's finish the calculations:

1 3 x y (x + y) (x 2 + 5) = 3 x y x 3 + 5 x + x 2 y + 5 y

Now let's see how to correctly divide one fraction by another. According to the rule, we need to replace this action by multiplying by the reciprocal fraction x 2 + 5 3 x x y:

1 x + y: 3 x y x 2 + 5 = 1 x + y x 2 + 5 3 x y

Let us reduce the resulting fraction to standard form:

1 x + y x 2 + 5 3 x y = 1 x 2 + 5 (x + y) 3 x y = x 2 + 5 3 x 2 y + 3 x y 2

Answer: 1 x + y 3 x y x 2 + 5 = 3 x y x 3 + 5 x + x 2 y + 5 y ; 1 x + y: 3 x y x 2 + 5 = x 2 + 5 3 x 2 y + 3 x y 2.

Quite often, the process of dividing and multiplying ordinary fractions produces results that can be abbreviated, for example, 2 9 · 3 8 = 6 72 = 1 12. When we do these things with algebraic fractions, we can also get reducible results. To do this, it is useful to first decompose the numerator and denominator of the original polynomial into separate factors. If necessary, reread the article on how to do this correctly. Let's look at an example of a problem in which you will need to reduce fractions.

Example 2

Condition: multiply the fractions x 2 + 2 · x + 1 18 · x 3 and 6 · x x 2 - 1 .

Solution

Before calculating the product, we factorize the numerator of the first original fraction and the denominator of the second. To do this, we need abbreviated multiplication formulas. We calculate:

x 2 + 2 x + 1 18 x 3 6 x x 2 - 1 = x + 1 2 18 x 3 6 x (x - 1) (x + 1) = x + 1 2 6 · x 18 · x 3 · x - 1 · x + 1

We have a fraction that can be reduced:

x + 1 2 6 x 18 x 3 x - 1 x + 1 = x + 1 3 x 2 (x - 1)

We wrote about how this is done in an article devoted to reducing algebraic fractions.

By multiplying the monomial and polynomial in the denominator, we get the result we need:

x + 1 3 x 2 (x - 1) = x + 1 3 x 3 - 3 x 2

Here is a transcript of the entire solution without explanation:

x 2 + 2 x + 1 18 x 3 6 x x 2 - 1 = x + 1 2 18 x 3 6 x (x - 1) (x + 1) = x + 1 2 6 · x 18 · x 3 · x - 1 · x + 1 = = x + 1 3 · x 2 · (x - 1) = x + 1 3 · x 3 - 3 · x 2

Answer: x 2 + 2 x + 1 18 x 3 6 x x 2 - 1 = x + 1 3 x 3 - 3 x 2 .

In some cases, it is convenient to transform the original fractions before multiplying or dividing to make further calculations faster and easier.

Example 3

Condition: divide 2 1 7 · x - 1 by 12 · x 7 - x .

Solution: Let's start by simplifying the algebraic fraction 2 1 7 · x - 1 to get rid of the fractional coefficient. To do this, we multiply both parts of the fraction by seven (this action is possible due to the main property of the algebraic fraction). As a result, we will get the following:

2 1 7 x - 1 = 7 2 7 1 7 x - 1 = 14 x - 7

We see that the denominator of the fraction 12 x 7 - x, by which we need to divide the first fraction, and the denominator of the resulting fraction are expressions opposite to each other. Changing the signs of the numerator and denominator 12 x 7 - x, we get 12 x 7 - x = - 12 x x - 7.

After all the transformations, we can finally move directly to dividing algebraic fractions:

2 1 7 x - 1: 12 x 7 - x = 14 x - 7: - 12 x x - 7 = 14 x - 7 x - 7 - 12 x = 14 x - 7 x - 7 - 12 x = = 14 - 12 x = 2 7 - 2 2 3 x = 7 - 6 x = - 7 6 x

Answer: 2 1 7 x - 1: 12 x 7 - x = - 7 6 x .

How to multiply or divide an algebraic fraction by a polynomial

To perform such an action, we can use the same rules that we gave above. First you need to represent the polynomial in the form of an algebraic fraction with one in the denominator. This operation is similar to converting a natural number into a fraction. For example, you can replace the polynomial x 2 + x − 4 on x 2 + x − 4 1. The resulting expressions will be identically equal.

Example 4

Condition: divide the algebraic fraction by the polynomial x + 4 5 · x · y: x 2 - 16.

Solution

x + 4 5 x y: x 2 - 16 = x + 4 5 x y: x 2 - 16 1 = x + 4 5 x y 1 x 2 - 16 = = x + 4 5 x y 1 (x - 4) x + 4 = (x + 4) 1 5 x y (x - 4) (x + 4) = 1 5 x y x - 4 = = 1 5 x 2 y - 20 x y

Answer: x + 4 5 x y: x 2 - 16 = 1 5 x 2 y - 20 x y.

If you notice an error in the text, please highlight it and press Ctrl+Enter

This lesson will cover the rules for multiplying and dividing algebraic fractions, as well as examples of how to apply these rules. Multiplying and dividing algebraic fractions is no different from multiplying and dividing ordinary fractions. At the same time, the presence of variables leads to somewhat more complex ways of simplifying the resulting expressions. Despite the fact that multiplying and dividing fractions is easier than adding and subtracting them, the study of this topic must be approached extremely responsibly, since there are many pitfalls in it that are usually not paid attention to. As part of the lesson, we will not only study the rules of multiplying and dividing fractions, but also analyze the nuances that may arise when applying them.

Subject:Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:Multiplying and dividing algebraic fractions

The rules for multiplying and dividing algebraic fractions are absolutely similar to the rules for multiplying and dividing ordinary fractions. Let's remind them:

That is, in order to multiply fractions, it is necessary to multiply their numerators (this will be the numerator of the product), and multiply their denominators (this will be the denominator of the product).

Division by a fraction is multiplication by an inverted fraction, that is, in order to divide two fractions, it is necessary to multiply the first of them (the dividend) by the inverted second (divisor).

Despite the simplicity of these rules, many people make mistakes in a number of special cases when solving examples on this topic. Let's take a closer look at these special cases:

In all these rules we used the following fact: .

Let's solve a few examples of multiplying and dividing ordinary fractions to remember how to use these rules.

Example 1

Note: When reducing fractions, we used the decomposition of numbers into prime factors. Let us remind you that prime numbers are those natural numbers that are divisible only by and by themselves. The remaining numbers are called composite . The number is neither prime nor composite. Examples of prime numbers: .

Example 2

Let us now consider one of the special cases with ordinary fractions.

Example 3

As you can see, multiplying and dividing ordinary fractions, if the rules are applied correctly, is not difficult.

Let's look at multiplication and division of algebraic fractions.

Example 4

Example 5

Note that it is possible and even necessary to reduce fractions after multiplication according to the same rules that we previously considered in the lessons devoted to reducing algebraic fractions. Let's look at a few simple examples for special cases.

Example 6

Example 7

Let's now look at some more complex examples of multiplying and dividing fractions.

Example 8

Example 9

Example 10

Example 11

Example 12

Example 13

Previously, we looked at fractions in which both the numerator and denominator were monomials. However, in some cases it is necessary to multiply or divide fractions whose numerators and denominators are polynomials. In this case, the rules remain the same, but to reduce it is necessary to use abbreviated multiplication formulas and bracketing.

Example 14

Example 15

Example 16

Example 17

Example 18

To perform multiplication of algebraic (rational) fractions, you need to:

1) Write the product of the numerators in the numerator, and write the product of the denominators of these fractions in the denominator.

In this case, polynomials are needed.

2) If possible, reduce the fraction.

Comment.

When multiplying, the sum and difference must be enclosed in parentheses.

Examples of multiplying algebraic fractions.

When multiplying algebraic fractions, we multiply the numerators separately, and the denominators of these fractions separately:

We reduce 36 and 45 by 9, 22 and 55 by 11, a² and by a a, b and b by b, c⁵ and c² by c²:

To multiply algebraic fractions, you multiply the numerator by the numerator and the denominator by the denominator. Since the numerators and denominators of these fractions contain polynomials, they are needed.

In the numerator of the first fraction, we take the common factor 3 out of brackets. We factor the numerator of the second fraction into factors as a difference of squares. The denominator of the first fraction is the square of the difference. In the denominator of the second fraction we take out the common factor 5:

The fraction can be reduced by (x+3) and (2x-1):

We multiply the numerator by the numerator, the denominator by the denominator. We factor the denominator of the second fraction using the difference of squares formula:

(a-b) and (b-a) differ only in sign. Let’s take the “minus” out of brackets, for example, in the numerator. After this, reduce the fraction by (a-b) and by a:

When multiplying algebraic fractions, we multiply the numerator by the numerator, and the denominator by the denominator. We try to factor the polynomials included in them.

In the first fraction, the numerator is the complete square of the sum, and the denominator is the sum of the cubes. In the second fraction in the numerator - (part of the formula for the sum of cubes), in the denominator there is a common factor of 3, which we put out of brackets:

We reduce the fraction by (x+3)² and (x²-3x+9):

In algebra, operations with algebraic (rational) fractions can occur both as a separate task and in the course of solving other examples, for example, solving equations and inequalities. That is why it is important to learn how to multiply, divide, add and subtract such fractions in time.

Example.

Find the product of algebraic fractions and .

Solution.

Before multiplying fractions, we factorize the polynomial in the numerator of the first fraction and the denominator of the second. The corresponding abbreviated multiplication formulas will help us with this: x 2 +2·x+1=(x+1) 2 and x 2 −1=(x−1)·(x+1) . Thus, .

Obviously, the resulting fraction can be reduced (we discussed this process in the article reducing algebraic fractions).

All that remains is to write the result in the form of an algebraic fraction, for which you need to multiply the monomial by the polynomial in the denominator: .

Usually the solution is written without explanation as a sequence of equalities:

Answer:

.

Sometimes with algebraic fractions that need to be multiplied or divided, you need to perform some transformations to make the operation easier and faster.

Example.

Divide an algebraic fraction by a fraction.

Solution.

Let's simplify the form of an algebraic fraction by getting rid of the fractional coefficient. To do this, we multiply its numerator and denominator by 7, which allows us to make the main property of an algebraic fraction, we have .

Now it has become clear that the denominator of the resulting fraction and the denominator of the fraction by which we need to divide are opposite expressions. Let's change the signs of the numerator and denominator of the fraction, we have .