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. From c³º what remains 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):
To understand how to reduce fractions, let's first look at an example.
To reduce a fraction means to divide the numerator and denominator by the same thing. Both 360 and 420 end in a digit, so we can reduce this fraction by 2. In the new fraction, both 180 and 210 are also divisible by 2, so we reduce this fraction by 2. In the numbers 90 and 105, the sum of the digits is divisible by 3, so both these numbers are divisible by 3, we reduce the fraction by 3. In the new fraction, 30 and 35 end in 0 and 5, which means both numbers are divisible by 5, so we reduce the fraction by 5. The resulting fraction of six-sevenths is irreducible. This is the final answer.
We can arrive at the same answer in a different way.
Both 360 and 420 end in zero, which means they are divisible by 10. We reduce the fraction by 10. In the new fraction, both the numerator 36 and the denominator 42 are divided by 2. We reduce the fraction by 2. In the next fraction, both the numerator 18 and the denominator 21 are divided by 3, which means we reduce the fraction by 3. We came to the result - six sevenths.
And one more solution.
Next time we'll look at examples of reducing fractions.
In this article we will look in detail at how reducing fractions. First, let's discuss what is called reducing a fraction. After this, let's talk about reducing a reducible fraction to an irreducible form. Next we will obtain the rule for reducing fractions and, finally, consider examples of the application of this rule.
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What does it mean to reduce a fraction?
We know that ordinary fractions are divided into reducible and irreducible fractions. You can guess from the names that reducible fractions can be reduced, but irreducible fractions cannot.
What does it mean to reduce a fraction? Reduce fraction- this means dividing its numerator and denominator by their positive and different from unity. It is clear that as a result of reducing a fraction, a new fraction is obtained with a smaller numerator and denominator, and, due to the basic property of the fraction, the resulting fraction is equal to the original one.
For example, let's reduce the common fraction 8/24 by dividing its numerator and denominator by 2. In other words, let's reduce the fraction 8/24 by 2. Since 8:2=4 and 24:2=12, this reduction results in the fraction 4/12, which is equal to the original fraction 8/24 (see equal and unequal fractions). As a result, we have .
Reducing ordinary fractions to irreducible form
Typically, the ultimate goal of reducing a fraction is to obtain an irreducible fraction that is equal to the original reducible fraction. This goal can be achieved by reducing the original reducible fraction into its numerator and denominator. As a result of such a reduction, an irreducible fraction is always obtained. Indeed, a fraction 
is irreducible, since it is known that 
And 
- . Here we will say that the greatest common divisor of the numerator and denominator of a fraction is the largest number by which this fraction can be reduced.
So, reducing a common fraction to an irreducible form consists of dividing the numerator and denominator of the original reducible fraction by their gcd.
Let's look at an example, for which we return to the fraction 8/24 and reduce it by the greatest common divisor of the numbers 8 and 24, which is equal to 8. Since 8:8=1 and 24:8=3, we come to the irreducible fraction 1/3. So, .
Note that the phrase “reduce a fraction” often means reducing the original fraction to its irreducible form. In other words, reducing a fraction very often refers to dividing the numerator and denominator by their greatest common factor (rather than by any common factor).
How to reduce a fraction? Rules and examples of reducing fractions
All that remains is to look at the rule for reducing fractions, which explains how to reduce a given fraction.
Rule for reducing fractions consists of two steps:
- firstly, the gcd of the numerator and denominator of the fraction is found;
- secondly, the numerator and denominator of the fraction are divided by their gcd, which gives an irreducible fraction equal to the original one.
Let's sort it out example of reducing a fraction according to the stated rule.
Example.
Reduce the fraction 182/195.
Solution.
Let's carry out both steps prescribed by the rule for reducing a fraction.
First we find GCD(182, 195) . It is most convenient to use the Euclid algorithm (see): 195=182·1+13, 182=13·14, that is, GCD(182, 195)=13.
Now we divide the numerator and denominator of the fraction 182/195 by 13, and we get the irreducible fraction 14/15, which is equal to the original fraction. This completes the reduction of the fraction.
Briefly, the solution can be written as follows: .
Answer:
This is where we can finish reducing fractions. But to complete the picture, let's look at two more ways to reduce fractions, which are usually used in easy cases.
Sometimes the numerator and denominator of the fraction being reduced is not difficult. Reducing a fraction in this case is very simple: you just need to remove all common factors from the numerator and denominator.
It is worth noting that this method follows directly from the rule of reducing fractions, since the product of all common prime factors of the numerator and denominator is equal to their greatest common divisor.
Let's look at the solution to the example.
Example.
Reduce the fraction 360/2 940.
Solution.
Let's factor the numerator and denominator into simple factors: 360=2·2·2·3·3·5 and 2,940=2·2·3·5·7·7. Thus, 
.
Now we get rid of the common factors in the numerator and denominator; for convenience, we simply cross them out: 
.
Finally, we multiply the remaining factors: , and the reduction of the fraction is completed.
Here is a short summary of the solution: 
.
Answer:
Let's consider another way to reduce a fraction, which consists of sequential reduction. Here, at each step, the fraction is reduced by some common divisor of the numerator and denominator, which is either obvious or easily determined using