Additional materials
Dear users, do not forget to leave your comments, reviews, wishes. All materials have been checked by an antivirus program.
Teaching aids and simulators in the Integral online store for grade 8
Algebra Electronic Workbook for Grade 8
Multimedia study guide for grade 8 "Algebra in 10 minutes"
Preliminary factorization of an algebraic fraction
Before starting work with fractions, namely multiplication and division, it is advisable to factorize the numerator and denominator. This will facilitate the factorization of the fraction that will result from the mathematical operation.For example, given a fraction:
$ \ frac (8x + 8y) (16) $.
We will produce identity transformation, that is, we factor out the numerator.
$ \ frac (8x + 8y) (16) = \ frac (8 (x + y)) (16) $.
Or, for example, a fraction like this is given:
$ \ frac (x ^ 2-y ^ 2) (x + 1) $.
It is better to bring it to this form:
$ \ frac (x ^ 2-y ^ 2) (x + 1) = \ frac ((x + y) (x-y)) (x + 1) $.
Don't forget about the property:
$ (b-a) ^ 2 = (a-b) ^ 2 $.
Multiplication of algebraic fractions with the same and different denominators
Multiplication of algebraic fractions is performed in the same way as multiplication of ordinary fractions. The numerators and denominators are multiplied among themselves.In the form of a formula, this can be represented as follows:
$ \ frac (a) (b) * \ frac (c) (d) = \ frac (ac) (bd) $
Let's look at a few examples.
Example 1.
Calculate:
$ \ frac (5x + 5y) (x-y) * \ frac (x ^ 2-y ^ 2) (10x) $.
Let us factor out the fraction.
$ \ frac (5x + 5y) (xy) * \ frac (x ^ 2-y ^ 2) (10x) = \ frac (5 (x + y)) (xy) * \ frac ((xy) (x + y)) (10x) $.
Let us bring both fractions to a common denominator (recall the lesson: "Adding and Subtracting Fractions", where there were tips on how to better and easier to select common denominator). As a result, we get a fraction.
$ \ frac (5 (x + y) (x-y) (x + y)) ((x-y) * 10x) = \ frac ((x + y) ^ 2) (2x) $
Example 2.
Calculate:
$ \ frac (7a ^ 3b ^ 5) (3a-3b) * \ frac (6b ^ 2-12ab + 6a ^ 2) (49a ^ 4b ^ 5) $.
Let's factorize it and cancel the fraction.
$ \ frac (7a ^ 3b ^ 5) (3a-3b) * \ frac (6 (b ^ 2-2ab + a ^ 2)) (49a ^ 4b ^ 5) = \ frac (7a ^ 3b ^ 5 * 6 (ba) ^ 2) (3 (ab) * 49a ^ 4b ^ 5) = \ frac (2 (ba) ^ 2) (7a (ab)) $.
Division of algebraic fractions with the same and different denominators
Division of fractions is performed in the same way as division of ordinary fractions, that is, you need to turn the "divisor" fraction and multiply.$ \ frac (a) (b): \ frac (c) (d) = \ frac (ad) (bc) $
Let's look at some examples.
Example 3.
Follow the steps:
$ \ frac (x ^ 3-1) (8y): \ frac (x ^ 2 + x + 1) (16y ^ 2) $.
Let us factor out the fractions.
$ \ frac (x ^ 3-1) (8y): \ frac (x ^ 2 + x + 1) (16y ^ 2) = \ frac ((x-1) (x ^ 2 + x + 1)) ( 8y): \ frac (x ^ 2 + x + 1) (16y ^ 2) $.
Now flip the fraction and multiply.
$ \ frac ((x-1) (x ^ 2 + x + 1) * 16y ^ 2) (8y * (x ^ 2 + x + 1)) = 2y * (x-1) $.
Example 4.
Calculate:
$ \ frac (a ^ 4-b ^ 4) (ab + 2b-3a-6): \ frac (b-a) (a + 2) $.
Factor and group the polynomials.
$ \ frac (a ^ 4-b ^ 4) (ab + 2b-3a-6): \ frac (ba) (a + 2) = \ frac ((a ^ 2-b ^ 2) (a ^ 2 + b ^ 2)) ((ab + 2b) - (3a + 6)): \ frac (ba) (a + 2) = $
$ \ frac ((a-b) (a + b) (a ^ 2 + b ^ 2)) (b (a + 2) -3 (a + 2)): \ frac (b-a) (a + 2) $.
Flip and multiply fractions.
$ \ frac ((ab) (a + b) (a ^ 2 + b ^ 2) (a + 2)) ((a + 2) (b-3) (ba)) = \ frac (- (a + b) (a ^ 2 + b ^ 2)) ((b-3)) $.
We are able to perform multiplication and division of arithmetic fractions, for example:
if the letters a, b, c, and d represent arithmetic integers.
The question arises whether these equalities are not valid if a, b, c and d denote: 1) some arithmetic numbers and 2) any relative numbers.
First of all, you will have to consider complex fractions, for example:
These examples are already enough to be convinced of the validity of the equalities related to multiplication and division of fractions, when the numbers a, b, c and d are any (whole or fractional) arithmetic. Note that there are only 2 basic equalities, namely:
It remains now to consider whether these equalities remain valid if some of the numbers a, b, c and d are assumed to be negative: if, for example, a is a negative number, b, c and d are positive, then the fraction is negative and the fraction is positive; therefore, for example, division by should result in a negative number, but we see that, according to our assumption, the expression should also express a negative number, that is, equality is justified in this case as well. It is also easy to consider other assumptions for the signs of a, b, c, and d. The result of this consideration is the conviction of the fairness of the equalities
and for the case when a, b, c and d express any relative numbers, that is, for multiplication and division of algebraic fractions, the same rules remain valid as for arithmetic ones.
Now we can perform multiplication and division of algebraic fractions. The greatest difficulty is presented here by the question of the reduction of fractions obtained after multiplication or division. If the algebraic fractions are single-term, then the reduction of the result obtained will not present difficulties, and if the fractions are algebraic, then it is necessary to first factor the numerator and denominator of each of these fractions into factors.
This lesson will consider the rules for multiplying and dividing algebraic fractions, as well as examples for applying these rules. Multiplication and division of algebraic fractions is no different from multiplication and division of ordinary fractions. At the same time, the presence of variables leads to somewhat more complex ways of simplifying the obtained 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 overlooked. As part of the lesson, we will not only study the rules for multiplying and dividing fractions, but also analyze the nuances that may arise when using them.
Subject:Algebraic fractions. Arithmetic operations on algebraic fractions
Lesson:Multiplication and division of algebraic fractions
The rules for multiplying and dividing algebraic ones are absolutely similar to the rules for multiplying and dividing ordinary fractions. Let's recall 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 fraction is multiplication by an inverted fraction, that is, in order to divide two fractions, you need to multiply the first of them (dividend) by the inverted second (divisor).
Despite the simplicity of these rules, many, when solving examples on this topic, make mistakes in a number of special cases. 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 multiplication and division of ordinary fractions in order to remember how to use these rules.
Example 1
Note: when reducing fractions, we used prime factorization. Recall that prime numbers
such natural numbers are called that are divisible only by and by itself. The rest of the numbers are called constituent
... The number is neither simple nor compound. Examples of primes: 
.
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, if the rules are applied correctly, is not difficult.
Consider the multiplication and division of algebraic fractions.
Example 4
Example 5
Note that it is possible and even necessary to cancel fractions after multiplication according to the same rules that we previously considered in the lessons on cancellation of algebraic fractions. Consider several simple examples for special cases.
Example 6
Example 7
Consider now a little more complex examples for multiplication and division of fractions.
Example 8
Example 9
Example 10
Example 11
Example 12
Example 13
Before that, we considered fractions in which both the numerator and the denominator were monomials. However, in some cases, it is necessary to multiply or divide fractions, the numerators and denominators of which are polynomials. In this case, the rules remain the same, but for reduction it is necessary to use the abbreviated multiplication formulas and parentheses.
Example 14
Example 15
Example 16
Example 17
Example 18
Example.
Find the product of algebraic fractions and.
Decision.
Before performing the multiplication of fractions, factor the polynomial in the numerator of the first fraction and the denominator of the second. The corresponding abbreviated multiplication formulas will help us in this: x 2 + 2 x + 1 = (x + 1) 2 and x 2 −1 = (x − 1) (x + 1). In this way, .
Obviously, the resulting fraction can be canceled 
(we discussed this process in the article cancellation of algebraic fractions).
It remains only to write the result in the form of an algebraic fraction, for which you need to multiply a monomial by a polynomial in the denominator: 
.
Usually, the solution is written without explanation in the form of a sequence of equalities: 
Answer:
.
Sometimes, with algebraic fractions that need to be multiplied or divided, you need to perform some transformations to make these steps easier and faster.
Example.
Divide an algebraic fraction by a fraction.
Decision.
Let us simplify the form of the algebraic fraction by getting rid of the fractional coefficient. To do this, multiply its numerator and denominator by 7, which allows us to make the main property of an algebraic fraction, we have 
.
Now it became clear that the denominator of the resulting fraction and the denominator of the fraction by which we need to divide are opposite expressions. We change the signs of the numerator and denominator of the fraction, we have 
.
In this article, we continue to explore the basic actions that can be performed with algebraic fractions. Here we will look at multiplication and division: first we derive the necessary rules, and then we illustrate them with problem solutions.
How to properly 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 common fraction on the other, we perform separately the multiplication of the numerators and separately the denominators, after which we write down the final fraction, placing the corresponding products in places. An example of such a calculation:
2 3 4 7 = 2 4 3 7 = 8 21
And when we need to divide common fractions, we do it by multiplying by the reciprocal of the divisor, for example:
2 3: 7 11 = 2 3 11 7 = 22 7 = 1 1 21
Multiplication and division of 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 with the product of the numerators in the numerator and the product of the denominators in the denominator.
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 definite polynomials, and b and d cannot be null.
Definition 2
In order to divide one algebraic fraction by another, you need to multiply the first fraction by the inverse 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 stand for polynomials, of which a, b, c and d cannot be null.
Let us dwell separately on what an inverse algebraic fraction is. It is a fraction that, when multiplied by the original, gives one in the end. That is, such fractions will be similar to mutually reciprocal numbers. Otherwise, we can say that an inverse algebraic fraction consists of the same values as the original one, but its numerator and denominator are reversed. So, in relation to the fraction a · b + 1 a 3, the fraction a 3 a · b + 1 will be inverse.
Solving problems on multiplication and division of algebraic fractions
In this paragraph, we will see how to correctly apply the rules outlined above in practice. Let's start with a simple and illustrative 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.
Decision
Let's do the multiplication first. According to the rule, you need to separately multiply the numerators and denominators:
1 x + y 3 x y x 2 + 5 = 1 3 x y (x + y) (x 2 + 5)
We got a new polynomial, which needs to be reduced to standard view... We 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 properly 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 y:
1 x + y: 3 x y x 2 + 5 = 1 x + y x 2 + 5 3 x y
Let us bring the resulting fraction to the 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, in the process of dividing and multiplying ordinary fractions, results are obtained that can be canceled, for example, 2 9 · 3 8 = 6 72 = 1 12. When we do this with algebraic fractions, we can also get canceled results. To do this, it is useful to first decompose the numerator and denominator of the original polynomial into separate factors. If necessary, re-read the article on how to do it correctly. Let's look at an example of a problem in which you need to perform fraction reduction.
Example 2
Condition: multiply the fractions x 2 + 2 x + 1 18 x 3 and 6 x x 2 - 1.
Decision
Before calculating the product, let's split the numerator of the first original fraction into separate factors and the denominator of the second. To do this, we need the abbreviated multiplication formulas. We calculate:
x 2 + 2 x + 1 18 x 3 6 xx 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've got 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 on cancellation of algebraic fractions.
Multiplying the monomial and the 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 record of the entire solution without explanation:
x 2 + 2 x + 1 18 x 3 6 xx 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 multiplication or division, so that further calculations become faster and easier.
Example 3
Condition: divide 2 1 7 x - 1 by 12 x 7 - x.
Solution: Start by simplifying the algebraic fraction 2 1 7 · x - 1 to get rid of the fractional coefficient. To do this, we multiply both sides of the fraction by seven (this action is possible due to the main property of the algebraic fraction). As a result, we 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 opposite expressions. 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 go directly to the division of algebraic fractions:
2 1 7 x - 1: 12 x 7 - x = 14 x - 7: - 12 xx - 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 have given above. First, you need to represent the polynomial as an algebraic fraction with a unit in the denominator. This action is similar to converting a natural number to a fraction. For example, you can replace the polynomial x 2 + x - 4 on the 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.
Decision
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 select it and press Ctrl + Enter