Algebraic multiplication. Multiplying algebraic fractions

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 multiplication 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.

In this article we will look at basic operations with algebraic fractions:

• reducing fractions
• multiplying fractions
• dividing fractions

Let's start with reduction of algebraic fractions.

It would seem that, algorithm obvious.

To reduce algebraic fractions, need to

1. Factor the numerator and denominator of the fraction.

2. Reduce equal factors.

However, schoolchildren often make the mistake of “reducing” not the factors, but the terms. For example, there are amateurs who “reduce” fractions by and get as a result , which, of course, is not true.

Let's look at examples:

1. Reduce fraction:

1. Let us factorize the numerator using the formula of the square of the sum, and the denominator using the formula of the difference of squares

2. Divide the numerator and denominator by

2. Reduce fraction:

1. Let's factorize the numerator. Since the numerator contains four terms, we use grouping.

2. Let's factorize the denominator. We can also use grouping.

3. Let's write down the fraction that we got and reduce the same factors:

Multiplying algebraic fractions.

When multiplying algebraic fractions, we multiply the numerator by the numerator, and multiply the denominator by the denominator.

Important! There is no need to rush to multiply the numerator and denominator of a fraction. After we have written down the product of the numerators of the fractions in the numerator, and the product of the denominators in the denominator, we need to factor each factor and reduce the fraction.

Let's look at examples:

3. Simplify the expression:

1. Let’s write the product of fractions: in the numerator the product of the numerators, and in the denominator the product of the denominators:

2. Let's factorize each bracket:

Now we need to reduce the same factors. Note that the expressions and differ only in sign: and as a result of dividing the first expression by the second we get -1.

So,

We divide algebraic fractions according to the following rule:

That is To divide by a fraction, you need to multiply by the "inverted" one.

We see that dividing fractions comes down to multiplying, and multiplication ultimately comes down to reducing fractions.

Let's look at an example:

4. Simplify the expression:

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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 slightly more in complex ways simplification of 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 using 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 these are called integers, which are divisible only by and by itself. The remaining numbers are called composite . The number is neither prime nor composite. Examples prime numbers: .

Example 2

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

Example 3

As you can see, multiplication and division of ordinary fractions, in the case correct application The rules are not complicated.

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 us now consider a little more complex examples on 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

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 .