What is the common denominator of fractions? Reducing fractions to a common denominator

26.09.2019

Initially I wanted to include cast methods common denominator in the section “Adding and subtracting fractions.” But there turned out to be so much information, and its importance is so great (after all, not only numerical fractions have common denominators), that it is better to study this issue separately.

So let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The basic property of a fraction comes to the rescue, which, let me remind you, sounds like this:

A fraction will not change if its numerator and denominator are multiplied by the same number other than zero.

Thus, if you choose the factors correctly, the denominators of the fractions will become equal - this process is called reduction to a common denominator. And the required numbers, “evening out” the denominators, are called additional factors.

Why do we need to reduce fractions to a common denominator? Here are just a few reasons:

Adding and subtracting fractions with different denominators. There is no other way to perform this operation;
Comparing fractions. Sometimes reduction to a common denominator greatly simplifies this task;
Solving problems involving fractions and percentages. Percentages are, in fact, ordinary expressions that contain fractions.

There are many ways to find numbers that, when multiplied by them, will make the denominators of fractions equal. We will consider only three of them - in order of increasing complexity and, in a sense, effectiveness.

Criss-cross multiplication

The simplest and reliable way, which is guaranteed to equalize the denominators. We will act “in a headlong manner”: we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:

As additional factors, consider the denominators of neighboring fractions. We get:

Yes, it's that simple. If you are just starting to study fractions, it is better to work using this method - this way you will insure yourself against many mistakes and are guaranteed to get the result.

The only drawback of this method is that you have to do a lot of counting, because the denominators are multiplied “over and over”, and the result can be very big numbers. This is the price to pay for reliability.

Common Divisor Method

This technique helps to significantly reduce calculations, but, unfortunately, it is used quite rarely. The method is as follows:

Before you go straight ahead (i.e., using the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided into the other.
The number resulting from this division will be an additional factor for the fraction with a smaller denominator.
In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is where the savings lie. At the same time, the probability of error is sharply reduced.

Task. Find the meanings of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divided without a remainder by the other, we use the method of common factors. We have:

Note that the second fraction was not multiplied by anything at all. In fact, we cut the amount of computation in half!

By the way, I didn’t take the fractions in this example by chance. If you're interested, try counting them using the criss-cross method. After reduction, the answers will be the same, but there will be much more work.

This is the power of the common divisors method, but, again, it can only be used when one of the denominators is divisible by the other without a remainder. Which happens quite rarely.

Reducing fractions to the lowest common denominator, rules, examples, solutions.

This article explains how to find the lowest common denominator And how to reduce fractions to a common denominator.

First, the definitions of common denominator of fractions and least common denominator are given, and it is shown how to find the common denominator of fractions. Below is a rule for reducing fractions to a common denominator and examples of the application of this rule are considered. In conclusion, examples of bringing three or more fractions to a common denominator are discussed.

What is called reducing fractions to a common denominator?

If common fractions have equal denominators, then these fractions are said to be reduced to a common denominator.

Thus, the fractions 45/76 and 143/76 are reduced to a common denominator of 76, and the fractions 1/3, 3/3, 17/3 and 1,000/3 are reduced to a common denominator of 3.

If the denominators of the fractions are not equal, then such fractions can always be reduced to a common denominator by multiplying their numerator and denominator by certain additional factors.

For example, ordinary fractions 2/5 and 7/4 with the help of additional factors 4 and 5, respectively, are reduced to a common denominator 20. Indeed, multiplying the numerator and denominator of the fraction 2/5 by 4, we obtain the fraction 8/20, and multiplying the numerator and denominator fractions 7/4 by 5, we arrive at the fraction 35/20 (see bringing fractions to a new denominator).

Now we can say what it is to reduce fractions to a common denominator. Reducing fractions to a common denominator- This is the multiplication of the numerators and denominators of given fractions by such additional factors that the result is fractions with the same denominators.

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Common denominator, definition, examples

Now it's time to define the common denominator of fractions.

In other words, the common denominator of a certain set of ordinary fractions is any natural number, which is divisible by all denominators of these fractions.

From the stated definition it follows that a given set of fractions has infinitely many common denominators, since there is an infinite number of common multiples of all denominators of the original set of fractions.

Positive common multiples of 4 and 6 are 12, 24, 36, 48, ... Any of these numbers is a common denominator of the fractions 1/4 and 5/6.

To consolidate the material, consider the solution to the following example.

Can the fractions 2/3, 23/6 and 7/12 be reduced to a common denominator of 150?

To answer the question posed, we need to find out whether the number 150 is a common multiple of the denominators 3, 6 and 12. To do this, we will check whether 150 is divisible by each of these numbers (if necessary, see the rules and examples of dividing natural numbers, as well as the rules and examples of dividing natural numbers with a remainder): 150:3=50, 150:6=25, 150:12=12 (rest.

So, 150 is not evenly divisible by 12, therefore 150 is not a common multiple of 3, 6 and 12. Therefore, the number 150 cannot be a common denominator of the original fractions.

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Lowest common denominator, how to find it?

In the set of numbers that are the common denominators of given fractions, there is a smallest natural number, which is called the least common denominator.

Let us formulate the definition of the lowest common denominator of these fractions.

It remains to deal with the question of how to find the least common divisor.

Since the least common multiple is the smallest positive common divisor of a given set of numbers, then the LCM of the denominators of the given fractions is the least common denominator of the given fractions.

Thus, finding the least common denominator of fractions comes down to finding the LCM of the denominators of these fractions.

Let's look at the solution to the example.

Find the lowest common denominator of the fractions 3/10 and 277/28.

The denominators of these fractions are 10 and 28. The desired least common denominator is found as the LCM of the numbers 10 and 28. In our case, it is easy to find the LCM by decomposing the numbers into prime factors: since 10=2·5, and 28=2·2·7, then LCM(15, 28)=2·2·5·7=140.

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How to reduce fractions to a common denominator? Rule, examples, solutions

Common fractions usually result in a lowest common denominator.

We will now write down a rule that explains how to reduce fractions to their lowest common denominator.

Rule for reducing fractions to lowest common denominator consists of three steps:

First, find the lowest common denominator of the fractions.
Second, an additional factor is calculated for each fraction by dividing the lowest common denominator by the denominator of each fraction.
Third, the numerator and denominator of each fraction are multiplied by its additional factor.

Let us apply the stated rule to solve the following example.

Reduce the fractions 5/14 and 7/18 to their lowest common denominator.

Let's perform all the steps of the algorithm for reducing fractions to the lowest common denominator.

First, we find the least common denominator, which is equal to the least common multiple of the numbers 14 and 18. Since 14=2·7 and 18=2·3·3, then LCM(14, 18)=2·3·3·7=126.

Now we calculate additional factors with the help of which the fractions 5/14 and 7/18 will be reduced to a denominator of 126. For the fraction 5/14 the additional factor is 126:14=9, and for the fraction 7/18 the additional factor is 126:18=7 .

It remains to multiply the numerators and denominators of the fractions 5/14 and 7/18 by additional factors 9 and 7, respectively.

We have And .

So, reducing the fractions 5/14 and 7/18 to the lowest common denominator is complete.

The resulting fractions were 45/126 and 49/126.

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Reducing three or more fractions to the lowest common denominator

The rule from the previous paragraph allows you to reduce not only two fractions, but also three fractions, and more of them, to the lowest common denominator.

Let's look at the example solution.

Reduce the four common fractions 3/2, 5/6, 3/8 and 17/18 to their lowest common denominator.

The least common denominator of these fractions is equal to the least common multiple of the numbers 2, 6, 8 and 18. To find the LCM(2, 6, 8, 18) we use the information from the section Finding the LCM of three or more numbers.

We get LCM(2, 6)=6, LCM(6, 8)=24, finally LCM(24, 18)=72, therefore LCM(2, 6, 8, 18)=72. So the lowest common denominator is 72.

Now we calculate additional factors. For the fraction 3/2 the additional factor is 72:2=36, for the fraction 5/6 it is 72:6=12, for the fraction 3/8 the additional factor is 72:8=9, and for the fraction 17/18 it is 72 :18=4.

Reducing fractions to a common denominator

There is one last step left in reducing the original fractions to the lowest common denominator: .

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Common denominator is any positive common multiple of all denominators of these fractions.

Lowest common denominator- This smallest number, from all common denominators of these fractions.

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics: textbook for 5th grade. educational institutions.
Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.

Common denominator of common fractions

If ordinary fractions have the same denominators, then these fractions have a common denominator. Eg,

they have a common denominator.

Common denominator This is a number that is the denominator for two or more regular fractions.

Fractions with different denominators can be reduced to a common denominator.

Providing fractions with a common denominator

Providing fractions with a common denominator Is replacing these fractions with different denominators the same fractions with the same denominators?

Fractions can simply be reduced to a common denominator or lowest common denominator.

Lowest common denominator This is the lowest common denominator of these fractions.

Common denominator of factions on the Internet

To give fractions the lowest common denominator, you need:

If possible, perform a fraction reduction.
Find the smallest common catalogs of these fractions. NOC will be their lowest common denominator.
Divide the LCM by the denominators of these fractions. This measure finds an additional factor for each of these fractions. Additional coefficient Is it a number that requires the members of a fraction to be multiplied to bring it to a common denominator?
Multiply the numerator and denominator of each fraction with additional factor.

Example.

1) Find the NOC names of these factions:

NOC(8, 12) = 24

2) Additional factors were found:

24: 8 = 3 (for ) and 24: 12 = 2 (for )

3) Multiply the members of each faction with an additional factor:

Decreasing the common denominator can be written in a shorter form by specifying an additional factor in addition to each fraction's counter (top right or top left) and not writing down the intermediate calculations:

The common denominator can be reduced more easily by multiplying the members of the first fraction with the second immanent share and the members of the second fraction with the denominator of the first.

Example. Get the common denominator of fractions and:

The product of their denominators can be taken as the common denominator of fractions.

Reducing fractions to a common denominator is used to add, subtract, and compare fractions with different denominators.

Reduction to common denominator calculator

This calculator will help you reduce common fractions to the lowest common denominator.

Just enter two fractions and click.

5.4.5. Examples of converting fractions to the lowest common denominator

The lowest common denominator of continued fractions is the lowest common denominator for those fractions. ( see section "Finding the least common multiple": 5.3.5. Find the least number of multiples (NOC) of the given numbers).

To reduce the fraction by the lowest common denominator, you need to: 1) find the least common multiple of the denominators of these fractions, and this will be the lowest common denominator.

2) finds an additional coefficient for each of the fractions, for which a new denominator is distributed with the name of each fraction. 3) multiply the numerator and denominator of each fraction with an additional factor.

Examples. To reduce the following fractions to the lowest common denominator.

We find the least common multi-digit denominator: LCM (5; 4) = 20, since 20 is the smallest number divided by 5 and 4.

For the first part, an additional coefficient of 4 (20 : 5 = 4). For the second fraction there is an additional coefficient of 5 (20 : 4 = 5). Multiply the number and denominator of the first fraction by 4, and the counter and denominator of the second fraction by 5.

20 ).

The lowest common denominator for these fractions is the number 8, since it is divisible by 4 and internally.

For the first fraction there is no additional factor (or we can say that it is equal to one), the second factor is an additional factor 2 (8 : 4 = 2). Multiply the numerator and denominator of the second fraction by 2.

Online calculator. Providing fractions with a common denominator

We have reduced these fractions to the lowest common denominator ( 8th place).

These factions are not intolerable.

The first faction was reduced by 4, and the second faction was reduced by 2. (See Examples for reducing common factions: Sitemap → 5.4.2.

Examples of reduction of common fractions). Finds NOC (16 ; 20) = 24· 5 = 16· 5 = 80. An additional factor for the 1st fraction is 5 (80 : 16 = 5). An additional factor for the second fraction is 4 (80 : 20 = 4).

We multiply the numerator and denominator of the first fraction with 5, and the counter and denominator of the second fraction with 4. The fractional information was given to the lowest common denominator ( 80 ).

Find the lowest common denominator of NOx (5 ; 6 and 15) = NOK (5 ; 6 and 15) = 30. An additional factor for the first fraction is 6 (30 : 5 = 6), is an additional factor in the second part of 5 (30 : 6 = 5), is an additional factor for the third fraction 2 (30 : 15 = 2).

The count and denominator of the first fraction are multiplied by 6, the count and denominator of the second fraction by 5, and the count and denominator of the third fraction by 2. The partial data was given the least common denominator 30 ).

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Lowest common denominator.

What is the lowest common denominator?

Definition:
Lowest common denominator is the smallest positive number that is a multiple of the denominators of these fractions.

How to reduce to the lowest common denominator? To answer this question, consider an example:

Reduce fractions with unlike denominators to their lowest common denominator.

Solution:
To find the lowest common denominator, you need to find the least common multiple (LCM) of the denominators of these fractions.

The first fraction has a denominator of 20; let’s factor it into prime factors.
20=2⋅5⋅2

Let us also decompose the second denominator of the fraction 14 into prime factors.
14=7⋅2

NOC(14,20)= 2⋅5⋅2⋅7=140

Answer: The lowest common denominator would be 140.

How to reduce a fraction to a common denominator?

You need to multiply the first fraction \(\frac(1)(20)\) by 7 to get a denominator of 140.

\(\frac(1)(20)=\frac(1 \times 7)(20 \times 7)=\frac(7)(140)\)
And multiply the second fraction by 10.

\(\frac(3)(14)=\frac(3 \times 10)(14 \times 10)=\frac(30)(140)\)

Rules or algorithm for reducing fractions to a common denominator.

Algorithm for reducing fractions to the lowest common denominator:

You need to factor the denominators of fractions into prime factors.
We need to find the least common multiple (LCM) for the denominators of these fractions.
Reduce fractions to a common denominator, that is, multiply both the numerator and denominator of the fraction by a factor.

Common denominator for several fractions.

How to find the common denominator for several fractions?

Let's look at an example:
Find the lowest common denominator for the fractions \(\frac(2)(11), \frac(1)(15), \frac(3)(22)\)

Solution:
Let's factor the denominators 11, 15 and 22 into prime factors.

The number 11 is already a simple number in itself, so there is no need to describe it.
Let's expand the number 15=5⋅3
Let's expand the number 22=11⋅2

Let's find the least common multiple (LCM) of the denominators 11, 15, and 22.
LCM(11, 15, 22)=11⋅2⋅5⋅3=330

We have found the lowest common denominator for these fractions. Now let's bring these fractions \(\frac(2)(11), \frac(1)(15), \frac(3)(22)\) to a common denominator equal to 330.

\(\begin(align)
\frac(2)(11)=\frac(2 \times 30)(11 \times 30)=\frac(60)(330) \\\\
\frac(1)(15)=\frac(1 \times 22)(15 \times 22)=\frac(22)(330) \\\\
\frac(3)(22)=\frac(3 \times 15)(22 \times 15)=\frac(60)(330) \\\\
\end(align)\)

I originally wanted to include common denominator techniques in the Adding and Subtracting Fractions section. But there turned out to be so much information, and its importance is so great (after all, not only numerical fractions have common denominators), that it is better to study this issue separately.

So, let's say we have two fractions with different denominators. And we want to make sure that the denominators become the same. The basic property of a fraction comes to the rescue, which, let me remind you, sounds like this:

A fraction will not change if its numerator and denominator are multiplied by the same number other than zero.

Why do we need to reduce fractions to a common denominator? Here are just a few reasons:

Adding and subtracting fractions with different denominators. There is no other way to perform this operation;
Comparing fractions. Sometimes reduction to a common denominator greatly simplifies this task;
Solving problems involving fractions and percentages. Percentages are essentially ordinary expressions that contain fractions.

Criss-cross multiplication

The simplest and most reliable method, which is guaranteed to equalize the denominators. We will act “in a headlong manner”: we multiply the first fraction by the denominator of the second fraction, and the second by the denominator of the first. As a result, the denominators of both fractions will become equal to the product of the original denominators. Take a look:

As additional factors, consider the denominators of neighboring fractions. We get:

The only drawback of this method is that you have to count a lot, because the denominators are multiplied “all the way”, and the result can be very large numbers. This is the price to pay for reliability.

Common Divisor Method

This technique helps to significantly reduce calculations, but, unfortunately, it is used quite rarely. The method is as follows:

Before you go straight ahead (i.e., using the criss-cross method), take a look at the denominators. Perhaps one of them (the one that is larger) is divided into the other.
The number resulting from this division will be an additional factor for the fraction with a smaller denominator.
In this case, a fraction with a large denominator does not need to be multiplied by anything at all - this is where the savings lie. At the same time, the probability of error is sharply reduced.

Task. Find the meanings of the expressions:

Note that 84: 21 = 4; 72: 12 = 6. Since in both cases one denominator is divided without a remainder by the other, we use the method of common factors. We have:

Note that the second fraction was not multiplied by anything at all. In fact, we cut the amount of computation in half!

This is the power of the common divisors method, but, again, it can only be used when one of the denominators is divisible by the other without a remainder. Which happens quite rarely.

Least common multiple method

When we reduce fractions to a common denominator, we are essentially trying to find a number that is divisible by each denominator. Then we bring the denominators of both fractions to this number.

There are a lot of such numbers, and the smallest of them will not necessarily be equal to the direct product of the denominators of the original fractions, as is assumed in the “criss-cross” method.

For example, for denominators 8 and 12, the number 24 is quite suitable, since 24: 8 = 3; 24: 12 = 2. This number is much less product 8 12 = 96.

The smallest number that is divisible by each of the denominators is called their least common multiple (LCM).

Notation: The least common multiple of a and b is denoted by LCM(a ; b) . For example, LCM(16, 24) = 48 ; LCM(8; 12) = 24 .

If you manage to find such a number, the total amount of calculations will be minimal. Look at the examples:

Task. Find the meanings of the expressions:

Note that 234 = 117 2; 351 = 117 3. Factors 2 and 3 are coprime (have no common factors other than 1), and factor 117 is common. Therefore LCM(234, 351) = 117 2 3 = 702.

Likewise, 15 = 5 3; 20 = 5 · 4. Factors 3 and 4 are coprime, and factor 5 is common. Therefore LCM(15, 20) = 5 3 4 = 60.

Now let's bring the fractions to common denominators:

Notice how useful it was to factorize the original denominators:

Having discovered identical factors, we immediately arrived at the least common multiple, which, generally speaking, is a non-trivial problem;
From the resulting expansion you can find out which factors are “missing” in each fraction. For example, 234 · 3 = 702, therefore, for the first fraction the additional factor is 3.

To appreciate how much of a difference the least common multiple method makes, try calculating these same examples using the criss-cross method. Of course, without a calculator. I think after this comments will be unnecessary.

Don't think that there won't be such complex fractions in the real examples. They meet all the time, and the above tasks are not the limit!

The only problem is how to find this very NOC. Sometimes everything can be found in a few seconds, literally “by eye,” but in general this is a complex computational task that requires separate consideration. We won't touch on that here.

This article explains how to find the lowest common denominator And how to reduce fractions to a common denominator. First, the definitions of common denominator of fractions and least common denominator are given, and it is shown how to find the common denominator of fractions. Below is a rule for reducing fractions to a common denominator and examples of the application of this rule are considered. In conclusion, examples of bringing three or more fractions to a common denominator are discussed.

Page navigation.

What is called reducing fractions to a common denominator?

Common denominator, definition, examples

Now it's time to define the common denominator of fractions.

In other words, the common denominator of a certain set of ordinary fractions is any natural number that is divisible by all the denominators of these fractions.

Determining the common denominator of fractions allows you to find the common denominators of given fractions. Let, for example, given the fractions 1/4 and 5/6, their denominators are 4 and 6, respectively. Positive common multiples of the numbers 4 and 6 are the numbers 12, 24, 36, 48, ... Any of these numbers is a common denominator of the fractions 1/4 and 5/6.

To consolidate the material, consider the solution to the following example.

Example.

Can the fractions 2/3, 23/6 and 7/12 be reduced to a common denominator of 150?

Solution.

To answer the question we need to find out whether the number 150 is a common multiple of the denominators 3, 6 and 12. To do this, let's check whether 150 is divisible by each of these numbers (if necessary, see rules and examples of dividing natural numbers, and rules and examples of dividing natural numbers with a remainder): 150:3=50, 150:6=25, 150:12=12 (remaining 6).

So, 150 is not evenly divisible by 12, therefore 150 is not a common multiple of 3, 6, and 12. Therefore, the number 150 cannot be the common denominator of the original fractions.

Answer:

It is forbidden.

Lowest common denominator, how to find it?

In the set of numbers that are common denominators of given fractions, there is smallest natural number, which is called the lowest common denominator. Let us formulate the definition of the lowest common denominator of these fractions.

Definition.

Lowest common denominator is the smallest number of all the common denominators of these fractions.

It remains to deal with the question of how to find the least common divisor.

Since is the least positive common divisor of a given set of numbers, the LCM of the denominators of the given fractions represents the least common denominator of the given fractions.

Thus, finding the lowest common denominator of fractions comes down to the denominators of those fractions. Let's look at the solution to the example.

Example.

Find the lowest common denominator of the fractions 3/10 and 277/28.

Solution.

The denominators of these fractions are 10 and 28. The desired lowest common denominator is found as the LCM of the numbers 10 and 28. In our case it’s easy: since 10=2·5, and 28=2·2·7, then LCM(15, 28)=2·2·5·7=140.

Answer:

140 .

How to reduce fractions to a common denominator? Rule, examples, solutions

Common fractions usually result in a lowest common denominator. We will now write down a rule that explains how to reduce fractions to their lowest common denominator.

Rule for reducing fractions to lowest common denominator consists of three steps:

First, find the lowest common denominator of the fractions.
Second, an additional factor is calculated for each fraction by dividing the lowest common denominator by the denominator of each fraction.
Third, the numerator and denominator of each fraction are multiplied by its additional factor.

Let us apply the stated rule to solve the following example.

Example.

Reduce the fractions 5/14 and 7/18 to their lowest common denominator.

Solution.

Let's perform all the steps of the algorithm for reducing fractions to the lowest common denominator.

First we find the least common denominator, which is equal to the least common multiple of the numbers 14 and 18. Since 14=2·7 and 18=2·3·3, then LCM(14, 18)=2·3·3·7=126.

Now we calculate additional factors with the help of which the fractions 5/14 and 7/18 will be reduced to the denominator 126. For the fraction 5/14 the additional factor is 126:14=9, and for the fraction 7/18 the additional factor is 126:18=7.

It remains to multiply the numerators and denominators of the fractions 5/14 and 7/18 by additional factors of 9 and 7, respectively. We have and .

So, reducing the fractions 5/14 and 7/18 to the lowest common denominator is complete. The resulting fractions were 45/126 and 49/126.

This method makes sense if the degree of the polynomial is not lower than two. In this case, the common factor can be not only a binomial of the first degree, but also of higher degrees.

To find a common factor terms of the polynomial, it is necessary to perform a number of transformations. The simplest binomial or monomial that can be taken out of brackets will be one of the roots of the polynomial. Obviously, in the case when the polynomial does not have a free term, there will be an unknown in the first degree - the polynomial, equal to 0.

More difficult to find a common factor is the case when the free term is not equal to zero. Then methods of simple selection or grouping are applicable. For example, let all the roots of the polynomial be rational, and all the coefficients of the polynomial are integers: y^4 + 3 y³ – y² – 9 y – 18.

Write down all the integer divisors of the free term. If a polynomial has rational roots, then they are among them. As a result of the selection, roots 2 and -3 are obtained. This means that the common factors of this polynomial will be the binomials (y - 2) and (y + 3).

The common factoring method is one of the components of factorization. The method described above is applicable if the coefficient of the highest degree is 1. If this is not the case, then a series of transformations must first be performed. For example: 2y³ + 19 y² + 41 y + 15.

Make a substitution of the form t = 2³·y³. To do this, multiply all the coefficients of the polynomial by 4: 2³·y³ + 19·2²·y² + 82·2·y + 60. After replacement: t³ + 19·t² + 82·t + 60. Now, to find the common factor, we apply the above method .

Besides, effective method Finding a common factor is the elements of a polynomial. It is especially useful when the first method does not, i.e. polynomial does not have rational roots. However, groupings are not always obvious. For example: The polynomial y^4 + 4 y³ – y² – 8 y – 2 has no integer roots.

Use grouping: y^4 + 4 y³ – y² – 8 y – 2 = y^4 + 4 y³ – 2 y² + y² – 8 y – 2 = (y^4 – 2 y²) + ( 4 y³ – 8 y) + y² – 2 = (y² - 2)*(y² + 4 y + 1). The common factor of the elements of this polynomial is (y² - 2).

Multiplication and division, just like addition and subtraction, are basic arithmetic operations. Without learning to solve examples of multiplication and division, a person will encounter many difficulties not only when studying more complex branches of mathematics, but even in the most ordinary everyday affairs. Multiplication and division are closely related, and the unknown components of examples and problems involving one of these operations are calculated using the other operation. At the same time, it is necessary to clearly understand that when solving examples, it makes absolutely no difference which objects you divide or multiply.

You will need

- multiplication table;
- calculator or sheet of paper and pencil.

Instructions

Write down the example you need. Label the unknown factor as an X. An example might look like this: a*x=b. Instead of the factor a and the product b in the example, there can be any or numbers. Remember the basic principle of multiplication: changing the places of the factors does not change the product. So unknown factor x can be placed absolutely anywhere.

To find the unknown factor in an example where there are only two factors, you just need to divide the product by the known factor. That is, this is done as follows: x=b/a. If you find it difficult to operate with abstract quantities, try to imagine this problem in the form of concrete objects. You, you have only apples and how many of them you will eat, but you don’t know how many apples everyone will get. For example, you have 5 family members, and there are 15 apples. Designate the number of apples intended for each as x. Then the equation will look like this: 5(apples)*x=15(apples). Unknown factor is found in the same way as in the equation with letters, that is, divide 15 apples among five family members, in the end it turns out that each of them ate 3 apples.

In the same way the unknown is found factor with the number of factors. For example, the example looks like a*b*c*x*=d. In theory, find with factor it is possible in the same way as in the later example: x=d/a*b*c. But the equation can be reduced to more simple view, denoting the product of known factors with another letter - for example, m. Find what m equals by multiplying numbers a,b and c: m=a*b*c. Then the whole example can be represented as m*x=d, and the unknown quantity will be equal to x=d/m.

If known factor and the product are fractions, the example is solved in exactly the same way as with . But in this case you need to remember the actions. When multiplying fractions, their numerators and denominators are multiplied. When dividing fractions, the numerator of the dividend is multiplied by the denominator of the divisor, and the denominator of the dividend is multiplied by the numerator of the divisor. That is, in this case the example will look like this: a/b*x=c/d. In order to find an unknown quantity, you need to divide the product by the known factor. That is, x=a/b:c/d =a*d/b*c.

Video on the topic

note

When solving examples with fractions, the fraction of a known factor can simply be reversed and the action performed as a multiplication of fractions.

A polynomial is the sum of monomials. A monomial is the product of several factors, which are a number or a letter. Degree unknown is the number of times it is multiplied by itself.

Instructions

Please provide it if it has not already been done. Similar monomials are monomials of the same type, that is, monomials with the same unknowns of the same degree.

Take, for example, the polynomial 2*y²*x³+4*y*x+5*x²+3-y²*x³+6*y²*y²-6*y²*y². This polynomial has two unknowns - x and y.

Connect similar monomials. Monomials with the second power of y and the third power of x will come to the form y²*x³, and monomials with the fourth power of y will cancel. It turns out y²*x³+4*y*x+5*x²+3-y²*x³.

Take y as the leading unknown letter. Find the maximum degree for unknown y. This is a monomial y²*x³ and, accordingly, degree 2.

Draw a conclusion. Degree polynomial 2*y²*x³+4*y*x+5*x²+3-y²*x³+6*y²*y²-6*y²*y² in x is equal to three, and in y is equal to two.

Find the degree polynomial√x+5*y by y. It is equal to the maximum degree of y, that is, one.

Find the degree polynomial√x+5*y in x. The unknown x is located, which means its degree will be a fraction. Since the root is a square root, the power of x is 1/2.

Draw a conclusion. For polynomial√x+5*y the x power is 1/2 and the y power is 1.

Video on the topic

Simplification algebraic expressions required in many areas of mathematics, including solving equations higher degrees, differentiation and integration. Several methods are used, including factorization. To apply this method, you need to find and make a general factor behind brackets.