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 .
In addition, an effective method for finding a common factor is the elements of a polynomial. It is especially useful when the first method does not, i.e. The polynomial has no 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 you can bring the equation to a simpler form by denoting the product of known factors with another letter - for example, m. Find what m equals by multiplying the 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 of algebraic expressions is required in many areas of mathematics, including solving higher-order equations, differentiation and integration. Several methods are used, including factorization. To apply this method, you need to find and make a general factor behind brackets.
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.
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What is called reducing fractions to a common 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.
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.
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.
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 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 (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 a smallest natural number, which is called the least 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 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.
The denominator of the arithmetic fraction a / b is the number b, which shows the size of the fractions of a unit from which the fraction is composed. The denominator of an algebraic fraction A / B is the algebraic expression B. To perform arithmetic operations with fractions, they must be reduced to the lowest common denominator.
You will need
- To work with algebraic fractions and find the lowest common denominator, you need to know how to factor polynomials.
Instructions
Let's consider reducing two arithmetic fractions n/m and s/t to the least common denominator, where n, m, s, t are integers. It is clear that these two fractions can be reduced to any denominator divisible by m and t. But they try to lead to the lowest common denominator. It is equal to the least common multiple of the denominators m and t of the given fractions. The least multiple (LMK) of a number is the smallest divisible by all given numbers at the same time. Those. in our case, we need to find the least common multiple of the numbers m and t. Denoted as LCM (m, t). Next, the fractions are multiplied by the corresponding ones: (n/m) * (LCM (m, t) / m), (s/t) * (LCM (m, t) / t).
Let's find the lowest common denominator of three fractions: 4/5, 7/8, 11/14. First, let's expand the denominators 5, 8, 14: 5 = 1 * 5, 8 = 2 * 2 * 2 = 2^3, 14 = 2 * 7. Next, calculate the LCM (5, 8, 14) by multiplying all the numbers included into at least one of the expansions. LCM (5, 8, 14) = 5 * 2^3 * 7 = 280. Note that if a factor occurs in the expansion of several numbers (factor 2 in the expansion of denominators 8 and 14), then we take the factor to a greater degree (2^3 in our case).
So, the general one is obtained. It is equal to 280 = 5 * 56 = 8 * 35 = 14 * 20. Here we get the numbers by which we need to multiply the fractions with the corresponding denominators in order to bring them to the lowest common denominator. We get 4/5 = 56 * (4/5) = 224/280, 7/8 = 35 * (7/8) = 245/280, 11/14 = 20 * (11/14) = 220/280.
Reduction of algebraic fractions to the lowest common denominator is carried out by analogy with arithmetic ones. For clarity, let's look at the problem using an example. Let two fractions (2 * x) / (9 * y^2 + 6 * y + 1) and (x^2 + 1) / (3 * y^2 + 4 * y + 1) be given. Let's factorize both denominators. Note that the denominator of the first fraction is a perfect square: 9 * y^2 + 6 * y + 1 = (3 * y + 1)^2. For
Content:
To add or subtract fractions with unlike denominators (the numbers below the fraction line), you first need to find their lowest common denominator (LCD). This number will be the smallest multiple that appears in the list of multiples of each denominator, that is, a number that is evenly divisible by each denominator. You can also calculate the least common multiple (LCM) of two or more denominators. In any case, we are talking about integers, the methods for finding which are very similar. Once you have determined the NOS, you can reduce fractions to a common denominator, which in turn allows you to add and subtract them.
Steps
1 Listing multiples
- 1
List the multiples of each denominator. Make a list of multiples of each denominator in the equation. Each list must consist of the product of the denominator by 1, 2, 3, 4, and so on.
- Example: 1/2 + 1/3 + 1/5
- Multiples of 2: 2 * 1 = 2; 2 * 2 = 4; 2 * 3 = 6; 2 * 4 = 8; 2 * 5 = 10; 2 * 6 = 12; 2 * 7 = 14; and so on.
- Multiples of 3: 3 * 1 = 3; 3 * 2 = 6; 3 *3 = 9; 3 * 4 = 12; 3 * 5 = 15; 3 * 6 = 18; 3 * 7 = 21; and so on.
- Multiples of 5: 5 * 1 = 5; 5 * 2 = 10; 5 * 3 = 15; 5 * 4 = 20; 5 * 5 = 25; 5 * 6 = 30; 5 * 7 = 35; and so on.
- 2
Determine the least common multiple. Go through each list and note any multiples that are common to all denominators. After identifying common multiples, determine the lowest denominator.
- Note that if a common denominator is not found, you may need to continue writing out multiples until a common multiple appears.
- It is better (and easier) to use this method when the denominators contain small numbers.
- In our example, the common multiple of all denominators is the number 30: 2 * 15 = 30 ; 3 * 10 = 30 ; 5 * 6 = 30
- NOZ = 30
- 3 In order to bring fractions to a common denominator without changing their meaning, multiply each numerator (the number above the fraction line) by a number equal to the quotient of NZ divided by the corresponding denominator.
- Example: (15/15) * (1/2); (10/10) * (1/3); (6/6) * (1/5)
- New equation: 15/30 + 10/30 + 6/30
- 4
Solve the resulting equation. After finding the NOS and changing the corresponding fractions, simply solve the resulting equation. Don't forget to simplify your answer (if possible).
- Example: 15/30 + 10/30 + 6/30 = 31/30 = 1 1/30
2 Using the greatest common divisor
- 1
List the divisors of each denominator. A divisor is an integer that divides a given number by a whole. For example, the divisors of the number 6 are the numbers 6, 3, 2, 1. The divisor of any number is 1, because any number is divisible by one.
- Example: 3/8 + 5/12
- Divisors 8: 1, 2, 4 , 8
- Divisors 12: 1, 2, 3, 4 , 6, 12
- 2
Find the greatest common divisor (GCD) of both denominators. After listing the factors of each denominator, note all the common factors. The greatest common factor is the largest common factor you will need to solve the problem.
- In our example, the common divisors for the denominators 8 and 12 are the numbers 1, 2, 4.
- GCD = 4.
- 3
Multiply the denominators together. If you want to use GCD to solve a problem, first multiply the denominators together.
- Example: 8 * 12 = 96
- 4
Divide the resulting value by GCD. Having received the result of multiplying the denominators, divide it by the gcd you calculated. The resulting number will be the lowest common denominator (LCD).
- Example: 96 / 4 = 24
- 5
- Example: 24 / 8 = 3; 24 / 12 = 2
- (3/3) * (3/8) = 9/24; (2/2) * (5/12) = 10/24
- 9/24 + 10/24
- 6
Solve the resulting equation.
- Example: 9/24 + 10/24 = 19/24
3 Factoring each denominator into prime factors
- 1
Factor each denominator into prime factors. Break down each denominator into prime factors, that is, prime numbers that, when multiplied, give the original denominator. Recall that prime factors are numbers that are divisible only by 1 or themselves.
- Example: 1/4 + 1/5 + 1/12
- Prime factors 4: 2 * 2
- Prime factors 5: 5
- Prime factors of 12: 2 * 2 * 3
- 2
Count the number of times each prime factor is present in each denominator. That is, determine how many times each prime factor appears in the list of factors of each denominator.
- Example: There are two 2 for denominator 4; zero 2 for 5; two 2 for 12
- There is a zero 3 for 4 and 5; one 3 for 12
- There is a zero 5 for 4 and 12; one 5 for 5
- 3
Take only the greatest number of times for each prime factor. Determine the greatest number of times each prime factor appears in any denominator.
- For example: the greatest number of times for a multiplier 2 - 2 times; For 3 - 1 time; For 5 - 1 time.
- 4
Write down the prime factors found in the previous step in order. Don't write down the number of times each prime factor appears in all the original denominators - do it based on the largest number of times (as described in the previous step).
- Example: 2, 2, 3, 5
- 5
Multiply these numbers. The result of the product of these numbers is equal to NOS.
- Example: 2 * 2 * 3 * 5 = 60
- NOZ = 60
- 6
Divide the NOZ by the original denominator. To calculate the multiplier required to reduce fractions to a common denominator, divide the NCD you found by the original denominator. Multiply the numerator and denominator of each fraction by this factor. You will get fractions with a common denominator.
- Example: 60/4 = 15; 60/5 = 12; 60/12 = 5
- 15 * (1/4) = 15/60; 12 * (1/5) = 12/60; 5 * (1/12) = 5/60
- 15/60 + 12/60 + 5/60
- 7
Solve the resulting equation. NOZ found; You can now add or subtract fractions. Don't forget to simplify your answer (if possible).
- Example: 15/60 + 12/60 + 5/60 = 32/60 = 8/15
4 Working with mixed numbers
- 1
Convert each mixed number to an improper fraction. To do this, multiply the whole part of the mixed number by the denominator and add it with the numerator - this will be the numerator of the improper fraction. Convert the whole number to a fraction too (just put 1 in the denominator).
- Example: 8 + 2 1/4 + 2/3
- 8 = 8/1
- 2 1/4, 2 * 4 + 1 = 8 + 1 = 9; 9/4
- Rewritten equation: 8/1 + 9/4 + 2/3
- 2
Find the lowest common denominator. Calculate the NVA using any method described in the previous sections. For this example, we will use the "listing multiples" method, in which multiples of each denominator are written down and the NOC is calculated based on them.
- Note that you don't need to list multiples for 1 , since any number multiplied by 1 , equal to itself; in other words, every number is a multiple of 1 .
- Example: 4 * 1 = 4; 4 * 2 = 8; 4 * 3 = 12 ; 4 * 4 = 16; etc.
- 3 * 1 = 3; 3 * 2 = 6; 3 * 3 = 9; 3 * 4 = 12 ; etc.
- NOZ = 12
- 3
Rewrite the original equation. Multiply the numerators and denominators of the original fractions by a number equal to the quotient of dividing the NZ by the corresponding denominator.
- For example: (12/12) * (8/1) = 96/12; (3/3) * (9/4) = 27/12; (4/4) * (2/3) = 8/12
- 96/12 + 27/12 + 8/12
- 4
Solve the equation. NOZ found; You can now add or subtract fractions. Don't forget to simplify your answer (if possible).
- Example: 96/12 + 27/12 + 8/12 = 131/12 = 10 11/12
What you will need
- Pencil
- Paper
- Calculator (optional)
In this lesson we will look at reducing fractions to a common denominator and solve problems on this topic. Let's define the concept of a common denominator and an additional factor, and remember about relatively prime numbers. Let's define the concept of the lowest common denominator (LCD) and solve a number of problems to find it.
Topic: Adding and subtracting fractions with different denominators
Lesson: Reducing fractions to a common denominator
Repetition. The main property of a fraction.
If the numerator and denominator of a fraction are multiplied or divided by the same natural number, you get an equal fraction.
For example, the numerator and denominator of a fraction can be divided by 2. We get the fraction. This operation is called fraction reduction. You can also perform the inverse transformation by multiplying the numerator and denominator of the fraction by 2. In this case, we say that we have reduced the fraction to a new denominator. The number 2 is called an additional factor.
Conclusion. A fraction can be reduced to any denominator that is a multiple of the denominator of the given fraction. To bring a fraction to a new denominator, its numerator and denominator are multiplied by an additional factor.
1. Reduce the fraction to the denominator 35.
The number 35 is a multiple of 7, that is, 35 is divisible by 7 without a remainder. This means that this transformation is possible. Let's find an additional factor. To do this, divide 35 by 7. We get 5. Multiply the numerator and denominator of the original fraction by 5.
2. Reduce the fraction to denominator 18.
Let's find an additional factor. To do this, divide the new denominator by the original one. We get 3. Multiply the numerator and denominator of this fraction by 3.
3. Reduce the fraction to a denominator of 60.
Dividing 60 by 15 gives an additional factor. It is equal to 4. Multiply the numerator and denominator by 4.
4. Reduce the fraction to the denominator 24
In simple cases, reduction to a new denominator is performed mentally. It is only customary to indicate the additional factor behind a bracket slightly to the right and above the original fraction.
A fraction can be reduced to a denominator of 15 and a fraction can be reduced to a denominator of 15. Fractions also have a common denominator of 15.
The common denominator of fractions can be any common multiple of their denominators. For simplicity, fractions are reduced to their lowest common denominator. It is equal to the least common multiple of the denominators of the given fractions.
Example. Reduce to the lowest common denominator of the fraction and .
First, let's find the least common multiple of the denominators of these fractions. This number is 12. Let's find an additional factor for the first and second fractions. To do this, divide 12 by 4 and 6. Three is an additional factor for the first fraction, and two is for the second. Let's bring the fractions to the denominator 12.
We brought the fractions to a common denominator, that is, we found equal fractions that have the same denominator.
Rule. To reduce fractions to their lowest common denominator, you must
First, find the least common multiple of the denominators of these fractions, it will be their least common denominator;
Secondly, divide the lowest common denominator by the denominators of these fractions, i.e. find an additional factor for each fraction.
Third, multiply the numerator and denominator of each fraction by its additional factor.
a) Reduce the fractions and to a common denominator.
The lowest common denominator is 12. The additional factor for the first fraction is 4, for the second - 3. We reduce the fractions to the denominator 24.
b) Reduce the fractions and to a common denominator.
The lowest common denominator is 45. Dividing 45 by 9 by 15 gives 5 and 3, respectively. We reduce the fractions to the denominator 45.
c) Reduce the fractions and to a common denominator.
The common denominator is 24. Additional factors are 2 and 3, respectively.
Sometimes it can be difficult to verbally find the least common multiple of the denominators of given fractions. Then the common denominator and additional factors are found using prime factorization.
Reduce the fractions and to a common denominator.
Let's factor the numbers 60 and 168 into prime factors. Let's write out the expansion of the number 60 and add the missing factors 2 and 7 from the second expansion. Let's multiply 60 by 14 and get a common denominator of 840. The additional factor for the first fraction is 14. The additional factor for the second fraction is 5. Let's bring the fractions to a common denominator of 840.
Bibliography
1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. - ZSh MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - ZSh MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O. and others. Mathematics: Textbook-interlocutor for 5-6 grades of secondary school. Math teacher's library. - Enlightenment, 1989.
You can download the books specified in clause 1.2. of this lesson.
Homework
Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S. and others. Mathematics 6. - M.: Mnemosyne, 2012. (link see 1.2)
Homework: No. 297, No. 298, No. 300.
Other tasks: No. 270, No. 290
