Addition and subtraction of algebraic fractions. Addition and subtraction of algebraic fractions Bringing algebraic fractions to a common denominator

This lesson will cover addition and subtraction. algebraic fractions with the same denominators. We already know how to add and subtract common fractions with the same denominators. It turns out that algebraic fractions follow the same rules. The ability to work with fractions with the same denominators is one of the cornerstones in learning the rules for working with algebraic fractions. In particular, understanding this topic will make it easy to master more difficult topic- addition and subtraction of fractions with different denominators. As part of the lesson, we will study the rules for adding and subtracting algebraic fractions with the same denominators, as well as analyze a number of typical examples

Rule for adding and subtracting algebraic fractions with the same denominators

Sfor-mu-li-ru-em pr-vi-lo slo-zhe-niya (you-chi-ta-niya) al-geb-ra-and-che-dro-bey with one-on-to-you -mi-know-on-te-la-mi (it is co-pa-yes-et with the ana-logic right-of-thumb for ordinary-but-ven-nyh-dr-bay): That is for the addition or you-chi-ta-niya al-geb-ra-and-che-dro-bey with one-to-you-mi-know-me-on-te-la-mi is necessary -ho-di-mo with-stand with-from-vet-stu-u-th al-geb-ra-i-che-sum of the number of-li-te-lei, and the sign-me-on-tel leave without iz-me-no-ny.

We will analyze this right-vi-lo both on the example of ordinary-but-vein-shot-beats, and on the example of al-geb-ra-and-che-dro- bey.

Examples of applying the rule for ordinary fractions

Example 1. Add fractions:.

Decision

Let's add the number-whether-they-whether draw-beat, and let's leave the sign-me-on-tel the same. After that, we divide the numer-li-tel and the sign-me-on-tel into simple multipliers and so-kra-tim. Let's get it: .

Note: standard error, I’ll start up something when resolving in a good kind of example, for -key-cha-et-sya in the following-du-u-sch-so-so-be-so-she-tion: . This is a gross mistake, since the sign-on-tel remains the same as it was in the original fractions.

Example 2. Add fractions:.

Decision

This za-da-cha is nothing from-whether-cha-et-sya from the previous one:.

Examples of applying the rule for algebraic fractions

From the usual-but-vein-nyh dro-bay per-rey-dem to al-geb-ra-i-che-skim.

Example 3. Add fractions:.

Solution: as already stated above, the addition of al-geb-ra-and-che-dro-bey is nothing from-is-cha-is-sya from the zhe-niya usually-but-vein-nyh dro-bay. Therefore, the solution method is the same:.

Example 4. You-honor fractions:.

Decision

You-chi-ta-nie al-geb-ra-and-che-dro-bey from-whether-cha-et-sya from the complication only by the fact that in the number of pi-sy-va-et-sya difference in the number of-li-te-lei is-run-nyh-dro-bay. So .

Example 5. You-honor fractions:.

Decision: .

Example 6. Simplify:.

Decision: .

Examples of applying the rule followed by reduction

In a fraction, someone-paradise is in a re-zul-ta-those addition or you-chi-ta-nia, it is possible to co-beautifully niya. In addition, you should not forget about the ODZ al-geb-ra-i-che-dro-bey.

Example 7. Simplify:.

Decision: .

Wherein . In general, if the ODZ of the out-of-hot-drow-bay owls-pa-yes-et with the ODZ of the total-go-howl, then you can not indicate it (after all, a fraction, in a lu-chen- naya in from-ve-those, also will not exist with co-from-vet-stu-u-s-knowing-che-no-yah-re-men-nyh). But if the ODZ is the source of the running dro-bay and from-ve-that does not co-pa-yes-et, then the ODZ indicates the need-ho-di-mo.

Example 8. Simplify:.

Decision: . At the same time, y (ODZ of the outgoing draw-bay does not coincide with the ODZ of re-zul-ta-ta).

Addition and subtraction of ordinary fractions with different denominators

To store and you-chi-tat al-geb-ra-and-che-fractions with different-we-know-me-on-te-la-mi, pro-ve-dem ana-lo -gyu from the usual-but-ven-ny-mi dro-bya-mi and re-re-not-sem it into al-geb-ra-and-che-fractions.

Ras-look at the simplest example for ordinary venous shots.

Example 1. Add fractions:.

Decision:

Let's remember the right-vi-lo-slo-drow-bay. For na-cha-la fractions, it is necessary to add-ve-sti to the common sign-me-to-te-lu. In the role of a general sign-me-on-te-la for ordinary-but-vein-draw-beats, you-stu-pa-et least common multiple(NOK) the source of the signs-me-on-the-lei.

Definition

The smallest-neck-to-tu-ral-number, someone-swarm is de-lit at the same time into numbers and.

To find the NOC, you need to de-lo-live know-me-on-the-whether into simple multipliers, and then choose to take everything pro- there are many, many, some of them are included in the difference between both signs-me-on-the-lei.

; . Then the LCM of numbers should include two twos and two threes:.

After finding the general sign-on-te-la, it is necessary for each of the dro-bays to find an additional multi- zhi-tel (fak-ti-che-ski, in de-pouring a common sign-me-on-tel on sign-me-on-tel co-from-rep-to-th-th fraction).

Then, each fraction is multiplied by a semi-chen-ny to-half-no-tel-ny multiplier. Fractions with the same-on-to-you-know-me-on-te-la-mi, warehouses and you-chi-tat someone we are on - studied in the past lessons.

By-lu-cha-eat: .

Answer:.

Ras-look-rim now the fold of al-geb-ra-and-che-dro-bey with different signs-me-on-te-la-mi. Sleep-cha-la, we-look at the fractions, know-me-on-the-whether some of them are-la-yut-sya number-la-mi.

Addition and subtraction of algebraic fractions with different denominators

Example 2. Add fractions:.

Decision:

Al-go-rhythm of re-she-niya ab-so-lyut-but ana-lo-gi-chen previous-du-sche-mu p-me-ru. It’s easy to take a common denominator on the given fractions: and add-to-full multipliers for each of them.

Answer:.

So, sfor-mu-li-ru-em al-go-rhythm of complication and you-chi-ta-niya al-geb-ra-and-che-dro-beats with different-we-know-me-on-te-la-mi:

1. Find the smallest common sign-me-on-tel draw-bay.

2. Find additional multipliers for each of the draw-bay fractions).

3. Do-multiply-live numbers-whether-the-whether on the co-ot-vet-stu-u-s-up to-half-no-tel-nye-multiple-those.

4. Add-to-live or you-honor the fractions, use the right-wi-la-mi of the fold and you-chi-ta-niya draw-bay with one-to-you-know -me-on-te-la-mi.

Ras-look-rim now an example with dro-bya-mi, in the know-me-on-the-le-there-are-there-are-there-are-beech-ven-nye you-ra-same -tion.

Frankly, these formulas should be remembered by any seventh grade student. It is simply impossible to study algebra even at the school level and not know the formula for the difference of squares or, say, the square of the sum. They are constantly encountered when simplifying algebraic expressions, when reducing fractions, and can even help in arithmetic calculations. Well, for example, you need to calculate in your mind: 3.16 2 - 2 3.16 1.16 + 1.16 2 . If you start counting it "on the forehead", it will turn out to be long and boring, and if you use the formula of the square of the difference, you will get the answer in 2 seconds!

So, seven formulas of "school" algebra that everyone should know:

Name	Formula
sum square	(A + B) 2 = A 2 + 2AB + B 2
The square of the difference	(A - B) 2 = A 2 - 2AB + B 2
Difference of squares	(A - B)(A + B) = A 2 - B 2
sum cube	(A + B) 3 = A 3 + 3A 2 B + 3AB 2 + B 3
difference cube	(A - B) 3 = A 3 - 3A 2 B + 3AB 2 - B 3
Sum of cubes	A 3 + B 3 = (A + B)(A 2 - AB + B 2)
Difference of cubes	A 3 - B 3 \u003d (A - B) (A 2 + AB + B 2)

Please note: there is no formula for the sum of squares! Don't let your imagination go too far.

What is the easiest way to remember all these formulas? Well, let's say, see certain analogies. For example, the formula for the square of the sum is similar to the formula for the square of the difference (the difference is only in one sign), and the formula for the cube of the sum is similar to the formula for the cube of the difference. Further, in the composition of the formulas for the difference of cubes and the sum of cubes, we see something similar to the square of the sum and the square of the difference (only the coefficient 2 is not enough).

But best of all these formulas (like any others!) are remembered in practice. Solve more examples to simplify algebraic expressions, and all the f-ly will be remembered by themselves.

Curious schoolchildren will probably be interested in summarizing the above facts. Here, let's say, there are formulas for the square and the cube of the sum. But what if we consider expressions like (A + B) 4 , (A + B) 5 , and even (A + B) n , where n is an arbitrary natural number? Can you see any pattern here?

Yes, such a pattern exists. An expression like (A + B) n is called Newton's binomial. I recommend inquisitive schoolchildren to derive the formulas for (A + B) 4 and (A + B) 5 themselves, and then try to see the general law: compare, for example, the degree of the corresponding binomial and the degree of each of the terms that are obtained by opening the brackets; compare the degree of the binomial with the number of terms; try to find patterns in the coefficients. We will not delve into this topic now (this requires a separate conversation!), But we will only write down the finished result:

(A + B) n = A n + C n 1 A n-1 B + C n 2 A n-2 B 2 + ... + C n k A n-k B k + ... + B n .

Here C n k = n!/(k! (n-k)!).

I remind you that n! is 1 2 ... n is the product of all natural numbers from 1 to n. This expression is called factorial of n. For example, 4! = 1 2 3 4 = 24. The factorial of zero is considered equal to one!

And what can be said about the difference of squares, the difference of cubes, etc.? Is there any pattern here? Is it possible to give a general formula for A n - B n ?

Yes, you can. Here is the formula:

A n - B n \u003d (A - B) (A n-1 + A n-2 B + A n-3 B 2 + ... + B n-1).

Moreover, for odd degrees n there is a similar formula for the sum:

A n + B n = (A + B)(A n-1 - A n-2 B + A n-3 B 2 - ... + B n-1).

We will not derive these formulas now (by the way, this is not very difficult), but it is certainly useful to know about their existence.

Ordinary fractions.

Addition of algebraic fractions

Remember!

You can only add fractions with the same denominators!

You can't add fractions without transformations

Can add fractions

When adding algebraic fractions with the same denominators:

the numerator of the first fraction is added to the numerator of the second fraction;
the denominator remains the same.

Consider an example of adding algebraic fractions.

Since the denominator of both fractions is “2a”, it means that the fractions can be added.

Add the numerator of the first fraction to the numerator of the second fraction, and leave the denominator the same. When adding fractions in the resulting numerator, we present similar ones.

Subtraction of algebraic fractions

When subtracting algebraic fractions with the same denominators:

the numerator of the second fraction is subtracted from the numerator of the first fraction.
the denominator remains the same.

Important!

Be sure to enclose the entire numerator of the subtracted fraction in brackets.

Otherwise, you will make a mistake in the signs when opening the brackets of the fraction to be subtracted.

Consider an example of subtracting algebraic fractions.

Since both algebraic fractions have a denominator " 2c", It means that these fractions can be subtracted.

Subtract from the numerator of the first fraction "(a + d)" the numerator of the second fraction "(a − b)". Do not forget to enclose the numerator of the subtracted fraction in brackets. When opening brackets, we use the rule of opening brackets.

Reduction of algebraic fractions to a common denominator

Let's consider another example. You need to add algebraic fractions.

You can't add fractions this way because they have different denominators.

Before adding algebraic fractions, they must bring to a common denominator.

The rules for reducing algebraic fractions to a common denominator are very similar to the rules for reducing ordinary fractions to a common denominator. .

As a result, we should get a polynomial that divides without a trace into each former denominator of the fractions.

To reduce algebraic fractions to a common denominator you need to do the following.

We work with numerical coefficients. We determine the LCM (least common multiple) for all numerical coefficients.
We work with polynomials. We define all different polynomials in the largest powers.
The product of the numerical coefficient and all the various polynomials to the highest degrees will be the common denominator.
Determine what each algebraic fraction needs to be multiplied by to get a common denominator.

Let's go back to our example.

Consider the denominators "15a" and "3" of both fractions and find a common denominator for them.

We work with numerical coefficients. We find the LCM (the least common multiple is the number that is divisible by each numerical coefficient without a remainder). For "15" and "3" - this is "15".
We work with polynomials. It is necessary to list all polynomials in the largest powers. The denominators "15a" and "5" have only
one monomial - "a".
We multiply the LCM from item 1 "15" and the monomial "a" from item 2. We will get " 15a». This will be the common denominator.
For each fraction, let's ask ourselves the question: "What do you need to multiply the denominator of this fraction to get" 15a "?".

Let's look at the first fraction. In this fraction, the denominator is already “ 15a”, which means that it does not need to be multiplied by anything.

Consider the second fraction. Let's ask the question: "What do you need to multiply" 3"To get" 15a"?" The answer is "5a".

When reducing the fraction to a common denominator, we multiply by " 5a" both numerator and denominator.

An abbreviated notation of bringing an algebraic fraction to a common denominator can be written through "houses".

To do this, keep the common denominator in mind. Above each fraction from above “in the house” we write what we multiply each of the fractions by.

Now that the fractions same denominators, fractions can be added.

Consider an example of subtracting fractions with different denominators.

Consider the denominators "(x − y)" and "(x + y)" of both fractions and find a common denominator for them.

We have two different polynomial in the denominators "(x − y)" and "(x + y)". Their product will be a common denominator, i.e. "(x − y)(x + y)" is the common denominator.

Adding and subtracting algebraic fractions using reduced multiplication formulas

In some examples, to reduce algebraic fractions to a common denominator, one must use the reduced multiplication formulas.

Consider an example of adding algebraic fractions, where we need to use the difference of squares formula.

In the first algebraic fraction, the denominator is "(p 2 − 36)". Obviously, the difference of squares formula can be applied to it.

After decomposing the polynomial "(p 2 − 36)" into the product of polynomials
“(p + 6)(p − 6)”, it can be seen that the polynomial “(p + 6)” is repeated in fractions. This means that the common denominator of the fractions will be the product of polynomials "(p + 6)(p − 6)".

Abbreviated expression formulas are very often used in practice, so it is advisable to learn them all by heart. Until this moment, we will serve faithfully, which we recommend printing out and keeping in front of our eyes all the time:

The first four formulas from the compiled table of abbreviated multiplication formulas allow you to square and cube the sum or difference of two expressions. The fifth is for briefly multiplying the difference and the sum of two expressions. And the sixth and seventh formulas are used to multiply the sum of two expressions a and b by their incomplete square of the difference (this is how the expression of the form a 2 −a b + b 2 is called) and the difference of two expressions a and b by the incomplete square of their sum (a 2 + a b+b 2 ) respectively.

It is worth noting separately that each equality in the table is an identity. This explains why abbreviated multiplication formulas are also called abbreviated multiplication identities.

When solving examples, especially in which the factorization of a polynomial takes place, FSU is often used in the form with the left and right parts rearranged:

The last three identities in the table have their own names. The formula a 2 −b 2 =(a−b) (a+b) is called difference of squares formula, a 3 +b 3 =(a+b) (a 2 −a b+b 2) - sum of cubes formula, a a 3 −b 3 =(a−b) (a 2 +a b+b 2) - cube difference formula. Please note that we did not name the corresponding formulas with rearranged parts from the previous FSU table.

Additional formulas

It does not hurt to add a few more identities to the table of abbreviated multiplication formulas.

Scopes of abbreviated multiplication formulas (FSU) and examples

The main purpose of the abbreviated multiplication formulas (FSU) is explained by their name, that is, it consists in a brief multiplication of expressions. However, the scope of the FSO is much wider, and is not limited to short multiplication. Let's list the main directions.

Undoubtedly, the central application of the reduced multiplication formula was found in performing identical transformations of expressions. Most often, these formulas are used in the process expression simplifications.

Example.

Simplify the expression 9·y−(1+3·y) 2 .

Decision.

AT given expression squaring can be done abbreviated, we have 9 y−(1+3 y) 2 =9 y−(1 2 +2 1 3 y+(3 y) 2). It remains only to open the brackets and give like terms: 9 y−(1 2 +2 1 3 y+(3 y) 2)= 9 y−1−6 y−9 y 2 =3 y−1−9 y 2.

In this article, we will look at basic operations with algebraic fractions:

fraction reduction
multiplication of fractions
division of fractions

Let's start with abbreviations of algebraic fractions.

Seemingly, algorithm obvious.

To reduce algebraic fractions, need

1. Factorize the numerator and denominator of a fraction.

2. Reduce the same multipliers.

However, schoolchildren often make the mistake of "reducing" not the factors, but the terms. For example, there are amateurs who "reduce" by in fractions and get as a result, which, of course, is not true.

Consider examples:

1. Reduce fraction:

1. We factorize the numerator according to the formula of the square of the sum, and the denominator according to the formula of the difference of squares

2. Divide the numerator and denominator by

2. Reduce fraction:

1. Factorize the numerator. Since the numerator contains four terms, we apply the grouping.

2. Factor the denominator. The same applies to grouping.

3. Let's write down the fraction that we got and reduce the same factors:

Multiplication of algebraic fractions.

When multiplying algebraic fractions, we multiply the numerator by the numerator, and we multiply the denominator by the denominator.

Important! No need to rush to perform multiplication in the numerator and denominator of a fraction. After we have written the product of the numerators of fractions in the numerator, and the product of the denominators in the denominator, we need to factor each factor and reduce the fraction.

Consider 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. We factorize each bracket:

Now we need to reduce the same multipliers. Note that the expressions and differ only in sign: and as a result of dividing the first expression by the second, we get -1.