Division and the numerator and denominator of the fraction on their common divisor, different from one, is called reducing a fraction.
To reduce a common fraction, you need to divide its numerator and denominator by the same natural number.
This number is the greatest common divisor of the numerator and denominator of the given fraction.
The following are possible decision recording forms Examples for reducing common fractions.
The student has the right to choose any form of recording.
Examples. Simplify fractions.
Reduce the fraction by 3 (divide the numerator by 3;
divide the denominator by 3).
Reduce the fraction by 7.
We perform the indicated actions in the numerator and denominator of the fraction.
The resulting fraction is reduced by 5.
Let's reduce this fraction 4)
on 5·7³- the greatest common divisor (GCD) of the numerator and denominator, which consists of the common factors of the numerator and denominator, taken to the power with the smallest exponent.
Let's factor the numerator and denominator of this fraction into prime factors.
We get: 756=2²·3³·7 And 1176=2³·3·7².
Determine the GCD (greatest common divisor) of the numerator and denominator of the fraction 5) .
This is the product of common factors taken with the lowest exponents.
GCD(756, 1176)= 2²·3·7.
We divide the numerator and denominator of this fraction by their gcd, i.e. by 2²·3·7 we get an irreducible fraction 9/14
.
Or it was possible to write the decomposition of the numerator and denominator in the form of a product of prime factors, without using the concept of power, and then reduce the fraction by crossing out the same factors in the numerator and denominator. When there are no identical factors left, we multiply the remaining factors separately in the numerator and separately in the denominator and write out the resulting fraction 9/14 .
And finally, it was possible to reduce this fraction 5) gradually, applying signs of dividing numbers to both the numerator and denominator of the fraction. We reason like this: numbers 756 And 1176 end in an even number, which means both are divisible by 2 . We reduce the fraction by 2 . The numerator and denominator of the new fraction are numbers 378 And 588 also divided into 2 . We reduce the fraction by 2 . We notice that the number 294 - even, and 189 is odd, and reduction by 2 is no longer possible. Let's check the divisibility of numbers 189 And 294 on 3 .
(1+8+9)=18 is divisible by 3 and (2+9+4)=15 is divisible by 3, hence the numbers themselves 189 And 294 are divided into 3 . We reduce the fraction by 3 . Next, 63 is divisible by 3 and 98 - No. Let's look at other prime factors. Both numbers are divisible by 7 . We reduce the fraction by 7 and we get the irreducible fraction 9/14 .
It is based on their basic property: if the numerator and denominator of a fraction are divided by the same non-zero polynomial, then an equal fraction will be obtained.
You can only reduce multipliers!
Members of polynomials cannot be abbreviated!
To reduce an algebraic fraction, the polynomials in the numerator and denominator must first be factorized.
Let's look at examples of reducing fractions.
The numerator and denominator of the fraction contain monomials. They represent work(numbers, variables and their powers), multipliers we can reduce.
We reduce the numbers by their greatest common divisor, that is, by the largest number by which each of these numbers is divided. For 24 and 36 this is 12. After reduction, 2 remains from 24, and 3 from 36.
We reduce the degrees by the degree with the lowest index. To reduce a fraction means to divide the numerator and denominator by the same divisor, and subtract the exponents.
a² and a⁷ are reduced to a². In this case, one remains in the numerator of a² (we write 1 only in the case when, after reduction, there are no other factors left. From 24, 2 remains, so we do not write 1 remaining from a²). From a⁷, after reduction, a⁵ remains.
b and b are reduced by b; the resulting units are not written.
c³º and c⁵ are shortened to c⁵. What remains from c³º is c²⁵, from c⁵ is one (we don’t write it). Thus,
The numerator and denominator of this algebraic fraction are polynomials. You cannot cancel terms of polynomials! (you cannot reduce, for example, 8x² and 2x!). To reduce this fraction, you need . The numerator has a common factor of 4x. Let's take it out of brackets:
Both the numerator and denominator have the same factor (2x-3). We reduce the fraction by this factor. In the numerator we got 4x, in the denominator - 1. According to 1 property of algebraic fractions, the fraction is equal to 4x.
You can only reduce factors (you cannot reduce this fraction by 25x²!). Therefore, the polynomials in the numerator and denominator of the fraction must be factorized.
The numerator is the complete square of the sum, the denominator is the difference of squares. After decomposition using abbreviated multiplication formulas, we obtain:
We reduce the fraction by (5x+1) (to do this, cross out the two in the numerator as an exponent, leaving (5x+1)² (5x+1)):
The numerator has a common factor of 2, let's take it out of brackets. The denominator is the formula for the difference of cubes:
As a result of the expansion, the numerator and denominator received the same factor (9+3a+a²). We reduce the fraction by it:
The polynomial in the numerator consists of 4 terms. the first term with the second, the third with the fourth, and remove the common factor x² from the first brackets. We decompose the denominator using the sum of cubes formula:
In the numerator, let’s take the common factor (x+2) out of brackets:
Reduce the fraction by (x+2):
Entry level
Converting Expressions. Detailed Theory (2019)
Converting Expressions
We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:
“It’s much simpler,” we say, but such an answer usually doesn’t work.
Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson, you yourself will simplify this example to (just!) an ordinary number (yes, to hell with these letters).
But before you start this lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, if you have not done this before, be sure to master the topics “” and “”.
Have you read it? If yes, then you are now ready.
Basic simplification operations
Now let's look at the basic techniques that are used to simplify expressions.
The simplest one is
1. Bringing similar
What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics. Similar are terms (monomials) with the same letter part. For example, in the sum, similar terms are and.
Do you remember?
To bring similar means to add several similar terms to each other and get one term.
How can we put the letters together? - you ask.
This is very easy to understand if you imagine that the letters are some kind of objects. For example, a letter is a chair. Then what is the expression equal to? Two chairs plus three chairs, how many will it be? That's right, chairs: .
Now try this expression: .
To avoid confusion, let different letters represent different objects. For example, - is (as usual) a chair, and - is a table. Then:
chairs tables chair tables chairs chairs tables
The numbers by which the letters in such terms are multiplied are called coefficients. For example, in a monomial the coefficient is equal. And in it is equal.
So, the rule for bringing similar ones is:
Examples:
Give similar ones:
Answers:
2. (and similar, since, therefore, these terms have the same letter part).
2. Factorization
This is usually the most important part in simplifying expressions. After you have given similar ones, most often the resulting expression needs to be factorized, that is, presented as a product. This is especially important in fractions: in order to be able to reduce a fraction, the numerator and denominator must be represented as a product.
You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned. To do this, decide a few examples(needs to be factorized):
Solutions:
3. Reducing a fraction.
Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?
That's the beauty of downsizing.
It's simple:
If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.
This rule follows from the basic property of a fraction:
That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).
To reduce a fraction you need:
1) numerator and denominator factorize
2) if the numerator and denominator contain common factors, they can be crossed out.
The principle, I think, is clear?
I would like to draw your attention to one typical mistake when abbreviating. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.
No abbreviations if the numerator or denominator is a sum.
For example: we need to simplify.
Some people do this: which is absolutely wrong.
Another example: reduce.
The “smartest” will do this: .
Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.
But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.
Here's another example: .
This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:
You can immediately divide it into:
To avoid such mistakes, remember an easy way to determine whether an expression is factorized:
The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation. That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized). If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).
To consolidate, solve a few yourself examples:
Answers:
1. I hope you didn’t immediately rush to cut and? It was still not enough to “reduce” units like this:
The first step should be factorization:
4. Adding and subtracting fractions. Reducing fractions to a common denominator.
Adding and subtracting ordinary fractions is a familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators. Let's remember:
Answers:
1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:
2. Here the common denominator is:
3. Here, first of all, we convert mixed fractions into improper ones, and then according to the usual scheme:
It's a completely different matter if the fractions contain letters, for example:
Let's start with something simple:
a) Denominators do not contain letters
Here everything is the same as with ordinary numerical fractions: we find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:
Now in the numerator you can give similar ones, if any, and factor them:
Try it yourself:
b) Denominators contain letters
Let's remember the principle of finding a common denominator without letters:
· first of all, we determine the common factors;
· then we write out all the common factors one at a time;
· and multiply them by all other non-common factors.
To determine the common factors of the denominators, we first factor them into prime factors:
Let us emphasize the common factors:
Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:
This is the common denominator.
Let's get back to the letters. The denominators are given in exactly the same way:
· factor the denominators;
· determine common (identical) factors;
· write out all common factors once;
· multiply them by all other non-common factors.
So, in order:
1) factor the denominators:
2) determine common (identical) factors:
3) write out all the common factors once and multiply them by all other (non-underlined) factors:
So there's a common denominator here. The first fraction must be multiplied by, the second - by:
By the way, there is one trick:
For example: .
We see the same factors in the denominators, only all with different indicators. The common denominator will be:
to a degree
to a degree
to a degree
to a degree.
Let's complicate the task:
How to make fractions have the same denominator?
Let's remember the basic property of a fraction:
Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!
See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?
So, another unshakable rule:
When you reduce fractions to a common denominator, use only the multiplication operation!
But what do you need to multiply by to get?
So multiply by. And multiply by:
We will call expressions that cannot be factorized “elementary factors.” For example, - this is an elementary factor. - Same. But no: it can be factorized.
What about the expression? Is it elementary?
No, because it can be factorized:
(you already read about factorization in the topic “”).
So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will deal with them in the same way.
We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).
The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:
Another example:
Solution:
Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:
Great! Then:
Another example:
Solution:
As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:
It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:
So let's write:
That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.
Now let's bring it to a common denominator:
Got it? Let's check it now.
Tasks for independent solution:
Answers:
Here we need to remember one more thing - the difference of cubes:
Please note that the denominator of the second fraction does not contain the formula “square of the sum”! The square of the sum would look like this: .
A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their double product. The partial square of the sum is one of the factors in the expansion of the difference of cubes:
What to do if there are already three fractions?
Yes, the same thing! First of all, let’s make sure that the maximum number of factors in the denominators is the same:
Please note: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction changes again to the opposite. As a result, it (the sign in front of the fraction) has not changed.
We write out the entire first denominator into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:
Hmm... It’s clear what to do with fractions. But what about the two?
It's simple: you know how to add fractions, right? So, we need to make two become a fraction! Let's remember: a fraction is a division operation (the numerator is divided by the denominator, in case you forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:
Just what you need!
5. Multiplication and division of fractions.
Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:
Procedure
What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:
Did you count?
It should work.
So, let me remind you.
The first step is to calculate the degree.
The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.
And finally, we perform addition and subtraction. Again, in any order.
But: the expression in parentheses is evaluated out of turn!
If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.
What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.
So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):
Okay, it's all simple.
But this is not the same as an expression with letters?
No, it's the same! Only instead of arithmetic operations, you need to do algebraic ones, that is, the actions described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.
Usually our goal is to represent the expression as a product or quotient.
For example:
Let's simplify the expression.
1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:
It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).
2) We get:
Multiplying fractions: what could be simpler.
3) Now you can shorten:
Well, that's all. Nothing complicated, right?
Another example:
Simplify the expression.
First, try to solve it yourself, and only then look at the solution.
First of all, let's determine the order of actions. First, let's add the fractions in parentheses, so instead of two fractions we get one. Then we will do division of fractions. Well, let's add the result with the last fraction. I will number the steps schematically:
Now I’ll show you the process, tinting the current action in red:
Finally, I will give you two useful tips:
1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.
2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.
Here are some tasks for you to solve on your own:
And what was promised at the very beginning:
Solutions (brief):
If you have coped with at least the first three examples, then consider yourself to have mastered the topic.
Now on to learning!
CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS
Basic simplification operations:
- Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
- Factorization: putting the common factor out of brackets, applying it, etc.
- Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
1) numerator and denominator factorize
2) if the numerator and denominator have common factors, they can be crossed out.IMPORTANT: only multipliers can be reduced!
- Adding and subtracting fractions:
; - Multiplying and dividing fractions:
;
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 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 a 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 a 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:
This article continues the topic of converting algebraic fractions: consider such an action as reducing algebraic fractions. Let's define the term itself, formulate a reduction rule and analyze practical examples.
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The meaning of reducing an algebraic fraction
In materials about common fractions, we looked at its reduction. We defined reducing a fraction as dividing its numerator and denominator by a common factor.
Reducing an algebraic fraction is a similar operation.
Definition 1
Reducing an algebraic fraction is the division of its numerator and denominator by a common factor. In this case, in contrast to the reduction of an ordinary fraction (the common denominator can only be a number), the common factor of the numerator and denominator of an algebraic fraction can be a polynomial, in particular, a monomial or a number.
For example, the algebraic fraction 3 x 2 + 6 x y 6 x 3 y + 12 x 2 y 2 can be reduced by the number 3, resulting in: x 2 + 2 x y 6 x 3 · y + 12 · x 2 · y 2 . We can reduce the same fraction by the variable x, and this will give us the expression 3 x + 6 y 6 x 2 y + 12 x y 2. It is also possible to reduce a given fraction by a monomial 3 x or any of the polynomials x + 2 y, 3 x + 6 y , x 2 + 2 x y or 3 x 2 + 6 x y.
The ultimate goal of reducing an algebraic fraction is a fraction of a simpler form, at best an irreducible fraction.
Are all algebraic fractions subject to reduction?
Again, from materials on ordinary fractions, we know that there are reducible and irreducible fractions. Irreducible fractions are fractions that do not have common numerator and denominator factors other than 1.
It’s the same with algebraic fractions: they may have common factors in the numerator and denominator, or they may not. The presence of common factors allows you to simplify the original fraction through reduction. When there are no common factors, it is impossible to optimize a given fraction using the reduction method.
In general cases, given the type of fraction it is quite difficult to understand whether it can be reduced. Of course, in some cases the presence of a common factor between the numerator and denominator is obvious. For example, in the algebraic fraction 3 x 2 3 y it is clear that the common factor is the number 3.
In the fraction - x · y 5 · x · y · z 3 we also immediately understand that it can be reduced by x, or y, or x · y. And yet, much more often there are examples of algebraic fractions, when the common factor of the numerator and denominator is not so easy to see, and even more often, it is simply absent.
For example, we can reduce the fraction x 3 - 1 x 2 - 1 by x - 1, while the specified common factor is not present in the entry. But the fraction x 3 - x 2 + x - 1 x 3 + x 2 + 4 · x + 4 cannot be reduced, since the numerator and denominator do not have a common factor.
Thus, the question of determining the reducibility of an algebraic fraction is not so simple, and it is often easier to work with a fraction of a given form than to try to find out whether it is reducible. In this case, such transformations take place that in particular cases make it possible to determine the common factor of the numerator and denominator or to draw a conclusion about the irreducibility of a fraction. We will examine this issue in detail in the next paragraph of the article.
Rule for reducing algebraic fractions
Rule for reducing algebraic fractions consists of two sequential actions:
- finding common factors of the numerator and denominator;
- if any are found, the action of reducing the fraction is carried out directly.
The most convenient method of finding common denominators is to factor the polynomials present in the numerator and denominator of a given algebraic fraction. This allows you to immediately clearly see the presence or absence of common factors.
The very action of reducing an algebraic fraction is based on the main property of an algebraic fraction, expressed by the equality undefined, where a, b, c are some polynomials, and b and c are non-zero. The first step is to reduce the fraction to the form a · c b · c, in which we immediately notice the common factor c. The second step is to perform a reduction, i.e. transition to a fraction of the form a b .
Typical examples
Despite some obviousness, let us clarify the special case when the numerator and denominator of an algebraic fraction are equal. Similar fractions are identically equal to 1 on the entire ODZ of the variables of this fraction:
5 5 = 1 ; - 2 3 - 2 3 = 1 ; x x = 1 ; - 3, 2 x 3 - 3, 2 x 3 = 1; 1 2 · x - x 2 · y 1 2 · x - x 2 · y ;
Since ordinary fractions are a special case of algebraic fractions, let us recall how they are reduced. The natural numbers written in the numerator and denominator are factored into prime factors, then the common factors are canceled (if any).
For example, 24 1260 = 2 2 2 3 2 2 3 3 5 7 = 2 3 5 7 = 2 105
The product of simple identical factors can be written as powers, and in the process of reducing a fraction, use the property of dividing powers with identical bases. Then the above solution would be:
24 1260 = 2 3 3 2 2 3 2 5 7 = 2 3 - 2 3 2 - 1 5 7 = 2 105
(numerator and denominator divided by a common factor 2 2 3). Or for clarity, based on the properties of multiplication and division, we give the solution the following form:
24 1260 = 2 3 3 2 2 3 2 5 7 = 2 3 2 2 3 3 2 1 5 7 = 2 1 1 3 1 35 = 2 105
By analogy, the reduction of algebraic fractions is carried out, in which the numerator and denominator have monomials with integer coefficients.
Example 1
The algebraic fraction is given - 27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z. It needs to be reduced.
Solution
It is possible to write the numerator and denominator of a given fraction as a product of simple factors and variables, and then carry out the reduction:
27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z = - 3 · 3 · 3 · a · a · a · a · a · b · b · c · z 2 · 3 · a · a · b · b · c · c · c · c · c · c · c · z = = - 3 · 3 · a · a · a 2 · c · c · c · c · c · c = - 9 a 3 2 c 6
However, a more rational way would be to write the solution as an expression with powers:
27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z = - 3 3 · a 5 · b 2 · c · z 2 · 3 · a 2 · b 2 · c 7 · z = - 3 3 2 · 3 · a 5 a 2 · b 2 b 2 · c c 7 · z z = = - 3 3 - 1 2 · a 5 - 2 1 · 1 · 1 c 7 - 1 · 1 = · - 3 2 · a 3 2 · c 6 = · - 9 · a 3 2 · c 6 .
Answer:- 27 a 5 b 2 c z 6 a 2 b 2 c 7 z = - 9 a 3 2 c 6
When the numerator and denominator of an algebraic fraction contain fractional numerical coefficients, there are two possible ways of further action: either divide these fractional coefficients separately, or first get rid of the fractional coefficients by multiplying the numerator and denominator by some natural number. The last transformation is carried out due to the basic property of an algebraic fraction (you can read about it in the article “Reducing an algebraic fraction to a new denominator”).
Example 2
The given fraction is 2 5 x 0, 3 x 3. It needs to be reduced.
Solution
It is possible to reduce the fraction this way:
2 5 x 0, 3 x 3 = 2 5 3 10 x x 3 = 4 3 1 x 2 = 4 3 x 2
Let's try to solve the problem differently, having first gotten rid of fractional coefficients - multiply the numerator and denominator by the least common multiple of the denominators of these coefficients, i.e. on LCM (5, 10) = 10. Then we get:
2 5 x 0, 3 x 3 = 10 2 5 x 10 0, 3 x 3 = 4 x 3 x 3 = 4 3 x 2.
Answer: 2 5 x 0, 3 x 3 = 4 3 x 2
When we reduce general algebraic fractions, in which the numerators and denominators can be either monomials or polynomials, there can be a problem where the common factor is not always immediately visible. Or moreover, it simply does not exist. Then, to determine the common factor or record the fact of its absence, the numerator and denominator of the algebraic fraction are factored.
Example 3
The rational fraction 2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 is given. It needs to be reduced.
Solution
Let us factor the polynomials in the numerator and denominator. Let's take it out of brackets:
2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49)
We see that the expression in parentheses can be converted using abbreviated multiplication formulas:
2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49) = 2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7)
It is clearly seen that it is possible to reduce a fraction by a common factor b 2 (a + 7). Let's make a reduction:
2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7) = 2 (a + 7) b (a - 7) = 2 a + 14 a b - 7 b
Let us write a short solution without explanation as a chain of equalities:
2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49) = = 2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7) = 2 (a + 7) b (a - 7) = 2 a + 14 a b - 7 b
Answer: 2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 a + 14 a b - 7 b.
It happens that common factors are hidden by numerical coefficients. Then, when reducing fractions, it is optimal to put the numerical factors at higher powers of the numerator and denominator out of brackets.
Example 4
Given the algebraic fraction 1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2 . It is necessary to reduce it if possible.
Solution
At first glance, the numerator and denominator do not have a common denominator. However, let's try to convert the given fraction. Let's take out the factor x in the numerator:
1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 = x 1 5 - 2 7 x 2 y 5 x 2 y - 3 1 2
Now you can see some similarity between the expression in brackets and the expression in the denominator due to x 2 y . Let us take out the numerical coefficients of the higher powers of these polynomials:
x 1 5 - 2 7 x 2 y 5 x 2 y - 3 1 2 = x - 2 7 - 7 2 1 5 + x 2 y 5 x 2 y - 1 5 3 1 2 = = - 2 7 x - 7 10 + x 2 y 5 x 2 y - 7 10
Now the common factor becomes visible, we carry out the reduction:
2 7 x - 7 10 + x 2 y 5 x 2 y - 7 10 = - 2 7 x 5 = - 2 35 x
Answer: 1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 = - 2 35 x .
Let us emphasize that the skill of reducing rational fractions depends on the ability to factor polynomials.
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