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Expression conversion. Detailed Theory (2019)

Expression conversion

Often we hear this unpleasant phrase: "simplify the expression." Usually, in this case, we have some kind of monster like this:

“Yes, much easier,” we say, but such an answer usually does not work.

Now I will teach you not to be afraid of any such tasks. Moreover, at the end of the lesson, you yourself will simplify this example to a (just!) ordinary number (yes, to hell with these letters).

But before you start this lesson, you need to be able to handle fractions and factor polynomials. Therefore, first, if you have not done this before, be sure to master the topics "" and "".

Read? If yes, then you are ready.

Basic simplification operations

Now we will analyze the main techniques that are used to simplify expressions.

The simplest of them is

1. Bringing similar

What are similar? You went through this in 7th grade, when letters first appeared in math instead of numbers. Similar are terms (monomials) with the same letter part. For example, in the sum, like terms are and.

Remembered?

To bring like terms means to add several similar terms to each other and get one term.

But how can we put letters together? - you ask.

This is very easy to understand if you imagine that the letters are some kind of objects. For example, the letter is a chair. Then what is the expression? Two chairs plus three chairs, how much will it be? That's right, chairs: .

Now try this expression:

In order not to get confused, let different letters denote different objects. For example, - this is (as usual) a chair, and - this is a table. Then:

chairs tables chair tables chairs chairs tables

The numbers by which the letters in such terms are multiplied are called coefficients. For example, in the monomial the coefficient is equal. And he is equal.

So, the rule for bringing similar:

Examples:

Bring similar:

Answers:

2. (and are similar, since, therefore, these terms have the same letter part).

2. Factorization

This is usually the most important part in simplifying expressions. After you have given similar ones, most often the resulting expression must be factored, that is, presented as a product. This is especially important in fractions: after all, in order to reduce a fraction, the numerator and denominator must be represented as a product.

You went through the detailed methods of factoring expressions in the topic "", so here you just have to remember what you have learned. To do this, solve a few examples(to be factored out):

Solutions:

3. Fraction reduction.

Well, what could be nicer than to cross out part of the numerator and denominator, and throw them out of your life?

That's the beauty of abbreviation.

It's simple:

If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.

This rule follows from the basic property of a fraction:

That is, the essence of the reduction operation is that We divide the numerator and denominator of a fraction by the same number (or by the same expression).

To reduce a fraction, you need:

1) numerator and denominator factorize

2) if the numerator and denominator contain common factors, they can be deleted.

The principle, I think, is clear?

I would like to draw your attention to one typical mistake in abbreviation. Although this topic is simple, but many people do everything wrong, not realizing that cut- this means divide numerator and denominator by the same number.

No abbreviations if the numerator or denominator is the sum.

For example: you need to simplify.

Some do this: which is absolutely wrong.

Another example: reduce.

"The smartest" will do this:.

Tell me what's wrong here? It would seem: - this is a multiplier, so you can reduce.

But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not decomposed into factors.

Here is another example: .

This expression is decomposed into factors, which means that you can reduce, that is, divide the numerator and denominator by, and then by:

You can immediately divide by:

To avoid such mistakes, remember an easy way to determine if an expression is factored:

The arithmetic operation that is performed last when calculating the value of the expression is the "main". That is, if you substitute some (any) numbers instead of letters, and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is decomposed into factors). If the last action is addition or subtraction, this means that the expression is not factored (and therefore cannot be reduced).

To fix it, solve it yourself a few examples:

Answers:

1. I hope you did not immediately rush to cut and? It was still not enough to “reduce” units like this:

The first step should be to factorize:

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

Addition and subtraction of ordinary fractions is a well-known operation: we look for a common denominator, multiply each fraction by the missing factor and add / subtract the numerators. Let's remember:

Answers:

1. The denominators and are coprime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. Here, first of all, we turn mixed fractions into improper ones, and then - according to the usual scheme:

It is quite another matter if the fractions contain letters, for example:

Let's start simple:

a) Denominators do not contain letters

Here everything is the same as with ordinary numerical fractions: we find a common denominator, multiply each fraction by the missing factor and add / subtract the numerators:

now in the numerator you can bring similar ones, if any, and factor them:

Try it yourself:

b) Denominators contain letters

Let's remember the principle of finding a common denominator without letters:

First of all, we determine the common factors;

Then we write out all the common factors once;

and multiply them by all other factors, not common ones.

To determine the common factors of the denominators, we first decompose them into simple factors:

We emphasize the common factors:

Now we write out the common factors once and add to them all non-common (not underlined) factors:

This is the common denominator.

Let's get back to the letters. The denominators are given in exactly the same way:

We decompose the denominators into factors;

determine common (identical) multipliers;

write out all the common factors once;

We multiply them by all other factors, not common ones.

So, in order:

1) decompose the denominators into factors:

2) determine the common (identical) factors:

3) write out all the common factors once and multiply them by all the other (not underlined) factors:

So the common denominator is here. The first fraction must be multiplied by, the second - by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to the extent

to the extent

to the extent

in degree.

Let's complicate the task:

How to make fractions have the same denominator?

Let's remember the basic property of a fraction:

Nowhere is it said that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What has been learned?

So, another unshakable rule:

When you bring fractions to a common denominator, use only the multiplication operation!

But what do you need to multiply to get?

Here on and multiply. And multiply by:

Expressions that cannot be factorized will be called "elementary factors". For example, is an elementary factor. - too. But - no: it is decomposed into factors.

What about expression? Is it elementary?

No, because it can be factorized:

(you already read about factorization in the topic "").

So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will do the same with them.

We see that both denominators have a factor. It will go to the common denominator in the power (remember why?).

The multiplier is elementary, and they do not have it in common, which means that the first fraction will simply have to be multiplied by it:

Another example:

Solution:

Before multiplying these denominators in a panic, you need to think about how to factor them? Both of them represent:

Excellent! Then:

Another example:

Solution:

As usual, we factorize the denominators. In the first denominator, we simply put it out of brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, they are already so similar ... And the truth is:

So let's write:

That is, it turned out like this: inside the bracket, we swapped the terms, and at the same time, the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now we bring to a common denominator:

Got it? Now let's check.

Tasks for independent solution:

Answers:

Here we must remember one more thing - the difference of cubes:

Please note that the denominator of the second fraction does not contain the formula "square of the sum"! The square of the sum would look like this:

A is the so-called incomplete square of the sum: the second term in it is the product of the first and last, and not their doubled product. The incomplete square of the sum is one of the factors in the expansion of the difference of cubes:

What if there are already three fractions?

Yes, the same! First of all, we will make sure that the maximum number of factors in the denominators is the same:

Pay attention: if you change the signs inside one bracket, the sign in front of the fraction changes to the opposite. When we change the signs in the second bracket, the sign in front of the fraction is reversed again. As a result, he (the sign in front of the fraction) has not changed.

We write out the first denominator in full in the common denominator, and then we add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it goes like this:

Hmm ... With fractions, it’s clear what to do. But what about the two?

It's simple: you know how to add fractions, right? So, you need to make sure that the deuce becomes a fraction! Remember: a fraction is a division operation (the numerator is divided by the denominator, in case you suddenly forgot). And there is nothing easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:

Exactly what is needed!

5. Multiplication and division of fractions.

Well, the hardest part is now over. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numeric expression? Remember, considering the value of such an expression:

Did you count?

It should work.

So, I remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, you can do them in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the parenthesized expression is evaluated out of order!

If several brackets are multiplied or divided by each other, we first evaluate the expression in each of the brackets, and then multiply or divide them.

What if there are other parentheses inside the brackets? Well, let's think: some expression is written inside the brackets. What is the first thing to do when evaluating an expression? That's right, calculate brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the order of actions for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But that's not the same as an expression with letters, is it?

No, it's the same! Only instead of arithmetic operations it is necessary to do algebraic operations, that is, the operations described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use it when working with fractions). Most often, for factorization, you need to use i or simply take the common factor out of brackets.

Usually our goal is to represent an expression as a product or quotient.

For example:

Let's simplify the expression.

1) First we simplify the expression in brackets. There we have the difference of fractions, and our goal is to represent it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression further, all factors here are elementary (do you still remember what this means?).

2) We get:

Multiplication of fractions: what could be easier.

3) Now you can shorten:

OK it's all over Now. Nothing complicated, right?

Another example:

Simplify the expression.

First, try to solve it yourself, and only then look at the solution.

First of all, let's define the procedure. First, let's add the fractions in brackets, instead of two fractions, one will turn out. Then we will do the division of fractions. Well, we add the result with the last fraction. I will schematically number the steps:

Now I will show the whole process, tinting the current action with red:

Finally, I will give you two useful tips:

1. If there are similar ones, they must be brought immediately. At whatever moment we have similar ones, it is advisable to bring them right away.

2. The same goes for reducing fractions: as soon as an opportunity arises to reduce, it must be used. The exception is fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And promised at the very beginning:

Solutions (brief):

If you coped with at least the first three examples, then you, consider, have mastered the topic.

Now on to learning!

EXPRESSION CONVERSION. SUMMARY AND BASIC FORMULA

Basic simplification operations:

• Bringing similar: to add (reduce) like terms, you need to add their coefficients and assign the letter part.
• Factorization: taking the common factor out of brackets, applying, etc.
• Fraction reduction: the numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, from which the value of the fraction does not change.
  1) numerator and denominator factorize
  2) if there are common factors in the numerator and denominator, they can be crossed out.

  IMPORTANT: only multipliers can be reduced!

• Addition and subtraction of fractions:
  ;
• Multiplication and division of fractions:
  ;

I. Expressions in which numbers, signs of arithmetic operations and brackets can be used along with letters are called algebraic expressions.

Examples of algebraic expressions:

2m-n; 3 · (2a+b); 0.24x; 0.3a-b · (4a + 2b); a 2 - 2ab;

Since a letter in an algebraic expression can be replaced by some different numbers, the letter is called a variable, and the algebraic expression itself is called an expression with a variable.

II. If in an algebraic expression letters (variables) are replaced by their values ​​and the specified actions are performed, then the resulting number is called the value of the algebraic expression.

Examples. Find the value of an expression:

1) a + 2b -c for a = -2; b = 10; c = -3.5.

2) |x| + |y| -|z| at x = -8; y=-5; z = 6.

Solution.

1) a + 2b -c for a = -2; b = 10; c = -3.5. Instead of variables, we substitute their values. We get:

— 2+ 2 · 10- (-3,5) = -2 + 20 +3,5 = 18 + 3,5 = 21,5.

2) |x| + |y| -|z| at x = -8; y=-5; z = 6. We substitute the indicated values. Remember that the modulus of a negative number is equal to its opposite number, and the modulus of a positive number is equal to this number itself. We get:

|-8| + |-5| -|6| = 8 + 5 -6 = 7.

III. The values ​​of a letter (variable) for which the algebraic expression makes sense are called valid values ​​of the letter (variable).

Examples. At what values ​​of the variable the expression does not make sense?

Solution. We know that it is impossible to divide by zero, therefore, each of these expressions will not make sense with the value of the letter (variable) that turns the denominator of the fraction to zero!

In example 1), this is the value a = 0. Indeed, if instead of a we substitute 0, then the number 6 will need to be divided by 0, but this cannot be done. Answer: expression 1) does not make sense when a = 0.

In example 2) the denominator x - 4 = 0 at x = 4, therefore, this value x = 4 and cannot be taken. Answer: expression 2) does not make sense for x = 4.

In example 3) the denominator is x + 2 = 0 for x = -2. Answer: expression 3) does not make sense at x = -2.

In example 4) the denominator is 5 -|x| = 0 for |x| = 5. And since |5| = 5 and |-5| \u003d 5, then you cannot take x \u003d 5 and x \u003d -5. Answer: expression 4) does not make sense for x = -5 and for x = 5.
IV. Two expressions are said to be identically equal if, for any admissible values ​​of the variables, the corresponding values ​​of these expressions are equal.

Example: 5 (a - b) and 5a - 5b are identical, since the equality 5 (a - b) = 5a - 5b will be true for any values ​​of a and b. Equality 5 (a - b) = 5a - 5b is an identity.

Identity is an equality that is valid for all admissible values ​​of the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, the distribution property.

The replacement of one expression by another, identically equal to it, is called an identical transformation or simply a transformation of an expression. Identical transformations of expressions with variables are performed based on the properties of operations on numbers.

Examples.

a) convert the expression to identically equal using the distributive property of multiplication:

1) 10 (1.2x + 2.3y); 2) 1.5 (a -2b + 4c); 3) a·(6m -2n + k).

Solution. Recall the distributive property (law) of multiplication:

(a+b) c=a c+b c(distributive law of multiplication with respect to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the results).
(a-b) c=a c-b c(distributive law of multiplication with respect to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply by this number reduced and subtracted separately and subtract the second from the first result).

1) 10 (1.2x + 2.3y) \u003d 10 1.2x + 10 2.3y \u003d 12x + 23y.

2) 1.5 (a -2b + 4c) = 1.5a -3b + 6c.

3) a (6m -2n + k) = 6am -2an +ak.

b) transform the expression into identically equal using the commutative and associative properties (laws) of addition:

4) x + 4.5 + 2x + 6.5; 5) (3a + 2.1) + 7.8; 6) 5.4s -3 -2.5 -2.3s.

Solution. We apply the laws (properties) of addition:

a+b=b+a(displacement: the sum does not change from the rearrangement of the terms).
(a+b)+c=a+(b+c)(associative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).

4) x + 4.5 + 2x + 6.5 = (x + 2x) + (4.5 + 6.5) = 3x + 11.

5) (3a + 2.1) + 7.8 = 3a + (2.1 + 7.8) = 3a + 9.9.

6) 6) 5.4s -3 -2.5 -2.3s = (5.4s -2.3s) + (-3 -2.5) = 3.1s -5.5.

in) transform the expression into identically equal using the commutative and associative properties (laws) of multiplication:

7) 4 · X · (-2,5); 8) -3,5 · 2y · (-one); 9) 3a · (-3) · 2s.

Solution. Let's apply the laws (properties) of multiplication:

a b=b a(displacement: permutation of factors does not change the product).
(a b) c=a (b c)(combinative: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third).

7) 4 · X · (-2,5) = -4 · 2,5 · x = -10x.

8) -3,5 · 2y · (-1) = 7y.

9) 3a · (-3) · 2s = -18as.

If an algebraic expression is given as a reducible fraction, then using the fraction reduction rule, it can be simplified, i.e. replace identically equal to it by a simpler expression.

Examples. Simplify by using fraction reduction.

Solution. To reduce a fraction means to divide its numerator and denominator by the same number (expression) other than zero. Fraction 10) will be reduced by 3b; fraction 11) reduce by a and fraction 12) reduce by 7n. We get:

Algebraic expressions are used to formulate formulas.

A formula is an algebraic expression written as an equality that expresses the relationship between two or more variables. Example: the path formula you know s=v t(s is the distance traveled, v is the speed, t is the time). Remember what other formulas you know.

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Often in tasks it is required to give a simplified answer. While both the simplified and non-simplistic answers are correct, your instructor may lower your grade if you don't simplify your answer. Moreover, a simplified mathematical expression is much easier to work with. Therefore, it is very important to learn how to simplify expressions.

Steps

Correct order of mathematical operations

1. Remember the correct order of doing math operations. When simplifying a mathematical expression, there is a certain order to follow, as some mathematical operations take precedence over others and must be done first (in fact, not following the correct order of operations will lead you to the wrong result). Remember the following order of mathematical operations: expression in brackets, exponentiation, multiplication, division, addition, subtraction.

   • Note that knowing the correct order of operations will allow you to simplify most of the simplest expressions, but to simplify a polynomial (an expression with a variable) you need to know special tricks (see the next section).

2. Start by solving the expression in parentheses. In mathematics, parentheses indicate that the enclosed expression must be evaluated first. Therefore, when simplifying any mathematical expression, start by solving the expression enclosed in brackets (it does not matter what operations you need to perform inside the brackets). But remember that when working with an expression enclosed in brackets, you should follow the order of operations, that is, the terms in brackets are first multiplied, divided, added, subtracted, and so on.

   • For example, let's simplify the expression 2x + 4(5 + 2) + 3 2 - (3 + 4/2). Here we start with the expressions in brackets: 5 + 2 = 7 and 3 + 4/2 = 3 + 2 = 5.
     • The expression in the second pair of brackets simplifies to 5 because 4/2 must be divided first (according to the correct order of operations). If you do not follow this order, then you will get the wrong answer: 3 + 4 = 7 and 7 ÷ 2 = 7/2.
   • If there is another pair of parentheses within the parentheses, start the simplification by solving the expression in the inner parentheses, and then move on to solving the expression in the outer parentheses.

3. Raise to a power. After solving the expressions in brackets, move on to raising to a power (remember that a power has an exponent and a base). Raise the corresponding expression (or number) to a power and substitute the result into the expression given to you.

   • In our example, the only expression (number) in the power is 3 2: 3 2 = 9. In the expression given to you, substitute 9 instead of 3 2 and you will get: 2x + 4(7) + 9 - 5.

4. Multiply. Remember that the multiplication operation can be denoted by the following symbols: "x", "∙" or "*". But if there are no symbols between a number and a variable (for example, 2x) or between a number and a number in brackets (for example, 4(7)) then this is also a multiplication operation.

   • In our example, there are two multiplication operations: 2x (two times x) and 4(7) (four times seven). We do not know the value of x, so we will leave the expression 2x as it is. 4(7) \u003d 4 x 7 \u003d 28. Now you can rewrite the expression given to you like this: 2x + 28 + 9 - 5.

5. Divide. Remember that the division operation can be denoted by the following symbols: "/", "÷" or "-" (you can see the last symbol in fractions). For example, 3/4 is three divided by four.

   • In our example, there is no more division because you already divided 4 by 2 (4/2) when solving the parenthesized expression. Therefore, you can move on to the next step. Remember that most expressions don't have all the math operations at once (only some of them).

6. Fold up. When adding terms of an expression, you can start with the outermost (left) term, or you can first add those terms that add up easily. For example, in the expression 49 + 29 + 51 +71, it is first easier to add 49 + 51 = 100, then 29 + 71 = 100, and finally 100 + 100 = 200. It is much more difficult to add like this: 49 + 29 = 78; 78 + 51 = 129; 129 + 71 = 200.

   • In our 2x + 28 + 9 + 5 example, there are two addition operations. Let's start with the most extreme (left) term: 2x + 28; you can't add 2x and 28 because you don't know the value of x. Therefore, add 28 + 9 = 37. Now the expression can be rewritten as follows: 2x + 37 - 5.

7. Subtract. This is the last operation in the correct order of math operations. At this stage, you can also add negative numbers, or you can do it at the stage of adding members - this will not affect the final result in any way.

   • In our example 2x + 37 - 5, there is only one subtraction operation: 37 - 5 = 32.

8. At this stage, having done all the mathematical operations, you should get a simplified expression. But if the expression given to you contains one or more variables, then remember that the member with the variable will remain as it is. Solving (rather than simplifying) an expression with a variable involves finding the value of that variable. Sometimes expressions with a variable can be simplified using special methods (see the next section).

   • In our example, the final answer is 2x + 32. You can't add two terms until you know the value of x. Once you know the value of the variable, you can easily simplify this binomial.

   Simplifying Complex Expressions

   1. Addition of similar members. Remember that you can only subtract and add similar terms, that is, terms with the same variable and the same exponent. For example, you can add 7x and 5x, but you can't add 7x and 5x 2 (because the exponents are different here).

      • This rule also applies to members with multiple variables. For example, you can add 2xy 2 and -3xy 2 , but you can't add 2xy 2 and -3x 2 y or 2xy 2 and -3y 2 .
      • Consider an example: x 2 + 3x + 6 - 8x. Here the like terms are 3x and 8x, so they can be added together. The simplified expression looks like this: x 2 - 5x + 6.

   2. Simplify the number. In such a fraction, both the numerator and the denominator contain numbers (without a variable). A numerical fraction is simplified in several ways. First, just divide the denominator by the numerator. Second, factor the numerator and denominator and cancel the same factors (because when you divide a number by itself, you get 1). In other words, if both the numerator and the denominator have the same factor, you can discard it and get a simplified fraction.

      • For example, consider the fraction 36/60. Using a calculator, divide 36 by 60 and get 0.6. But you can simplify this fraction in another way by factoring the numerator and denominator: 36/60 = (6x6)/(6x10) = (6/6)*(6/10). Since 6/6 \u003d 1, then the simplified fraction: 1 x 6/10 \u003d 6/10. But this fraction can also be simplified: 6/10 \u003d (2x3) / (2 * 5) \u003d (2/2) * (3/5) \u003d 3/5.

   3. If the fraction contains a variable, you can reduce the same factors with the variable. Factor both the numerator and denominator and cancel the same factors even if they contain a variable (remember that here the same factors may or may not contain a variable).

      • Consider an example: (3x 2 + 3x)/(-3x 2 + 15x). This expression can be rewritten (factored) as: (x + 1)(3x)/(3x)(5 - x). Since the 3x term is in both the numerator and the denominator, it can be reduced to give you a simplified expression: (x + 1)/(5 - x). Consider another example: (2x 2 + 4x + 6)/2 = (2(x 2 + 2x + 3))/2 = x 2 + 2x + 3.
      • Please note that you cannot cancel any terms - only the same factors that are present in both the numerator and the denominator are cancelled. For example, in the expression (x(x + 2))/x, the variable (multiplier) "x" is in both the numerator and the denominator, so "x" can be reduced and get a simplified expression: (x + 2) / 1 \u003d x + 2. However, in the expression (x + 2)/x, the variable "x" cannot be reduced (because in the numerator "x" is not a factor).

   4. Open parenthesis. To do this, multiply the term outside the bracket by each term in the brackets. Sometimes it helps to simplify a complex expression. This applies to both members that are prime numbers and members that contain a variable.

      • For example, 3(x 2 + 8) = 3x 2 + 24 and 3x(x 2 + 8) = 3x 3 + 24x.
      • Please note that in fractional expressions, parentheses do not need to be opened if both the numerator and the denominator contain the same factor. For example, in the expression (3(x 2 + 8)) / 3x, you do not need to expand the brackets, since here you can reduce the factor 3 and get a simplified expression (x 2 + 8) / x. This expression is easier to work with; if you expanded the brackets, you would get the following complex expression: (3x 3 + 24x)/3x.

   5. Factorize the polynomials. Using this method, you can simplify some expressions and polynomials. Factoring is the opposite of parenthesis expansion, that is, an expression is written as a product of two expressions, each of which is enclosed in parentheses. In some cases, factoring allows you to shorten the same expression. In special cases (usually with quadratic equations), factoring will allow you to solve the equation.

      • Consider the expression x 2 - 5x + 6. It is decomposed into factors: (x - 3) (x - 2). Thus, if, for example, an expression is given (x 2 - 5x + 6)/(2(x - 2)), then you can rewrite it as (x - 3)(x - 2)/(2(x - 2)), reduce the expression (x - 2) and get a simplified expression (x - 3) / 2.
      • Factoring polynomials is used to solve (find roots) equations (an equation is a polynomial equated to 0). For example, consider the equation x 2 - 5x + 6 \u003d 0. Factoring it out, you get (x - 3) (x - 2) \u003d 0. Since any expression multiplied by 0 is 0, we can write it like this : x - 3 \u003d 0 and x - 2 \u003d 0. Thus, x \u003d 3 and x \u003d 2, that is, you found two roots of the equation given to you.

Simplifying algebraic expressions is one of the keys to learning algebra and an extremely useful skill for all mathematicians. Simplification allows you to reduce a complex or long expression to a simple expression that is easy to work with. Basic simplification skills are good even for those who are not enthusiastic about mathematics. By following a few simple rules, many of the most common types of algebraic expressions can be simplified without any special mathematical knowledge.

Steps

Important Definitions

1. Similar members. These are members with a variable of the same order, members with the same variables, or free members (members that do not contain a variable). In other words, like terms include one variable to the same extent, include several identical variables, or do not include a variable at all. The order of the terms in the expression does not matter.

   • For example, 3x 2 and 4x 2 are like terms because they contain the variable "x" of the second order (in the second power). However, x and x 2 are not similar members, since they contain the variable "x" of different orders (first and second). Similarly, -3yx and 5xz are not similar members because they contain different variables.

2. Factorization. This is finding such numbers, the product of which leads to the original number. Any original number can have several factors. For example, the number 12 can be decomposed into the following series of factors: 1 × 12, 2 × 6 and 3 × 4, so we can say that the numbers 1, 2, 3, 4, 6 and 12 are factors of the number 12. The factors are the same as divisors , that is, the numbers by which the original number is divisible.

   • For example, if you want to factor the number 20, write it like this: 4×5.
   • Note that when factoring, the variable is taken into account. For example, 20x = 4(5x).
   • Prime numbers cannot be factored because they are only divisible by themselves and 1.

3. Remember and follow the order of operations to avoid mistakes.

   • Parentheses
   • Degree
   • Multiplication
   • Division
   • Addition
   • Subtraction

   Casting Like Members

   1. Write down the expression. The simplest algebraic expressions (which do not contain fractions, roots, and so on) can be solved (simplified) in just a few steps.

      • For example, simplify the expression 1 + 2x - 3 + 4x.

   2. Define similar members (members with a variable of the same order, members with the same variables, or free members).

      • Find similar terms in this expression. The terms 2x and 4x contain a variable of the same order (first). Also, 1 and -3 are free members (do not contain a variable). Thus, in this expression, the terms 2x and 4x are similar, and the members 1 and -3 are also similar.

   3. Give similar terms. This means adding or subtracting them and simplifying the expression.

      • 2x+4x= 6x
      • 1 - 3 = -2

   4. Rewrite the expression taking into account the given terms. You will get a simple expression with fewer terms. The new expression is equal to the original.

      • In our example: 1 + 2x - 3 + 4x = 6x - 2, that is, the original expression is simplified and easier to work with.

   5. Observe the order in which operations are performed when casting like terms. In our example, it was easy to bring similar terms. However, in the case of complex expressions in which members are enclosed in brackets and fractions and roots are present, it is not so easy to bring such terms. In these cases, follow the order of operations.

      • For example, consider the expression 5(3x - 1) + x((2x)/(2)) + 8 - 3x. Here it would be a mistake to immediately define 3x and 2x as like terms and quote them, because first you need to expand the parentheses. Therefore, perform the operations in their order.
        • 5(3x-1) + x((2x)/(2)) + 8 - 3x
        • 15x - 5 + x(x) + 8 - 3x
        • 15x - 5 + x 2 + 8 - 3x. Now, when the expression contains only addition and subtraction operations, you can cast like terms.
        • x 2 + (15x - 3x) + (8 - 5)
        • x 2 + 12x + 3

   Parenthesizing the multiplier

   1. Find the greatest common divisor (gcd) of all coefficients of the expression. GCD is the largest number by which all coefficients of the expression are divisible.

      • For example, consider the equation 9x 2 + 27x - 3. In this case, gcd=3, since any coefficient of this expression is divisible by 3.

   2. Divide each term of the expression by gcd. The resulting terms will contain smaller coefficients than in the original expression.

      • In our example, divide each expression term by 3.
        • 9x2/3=3x2
        • 27x/3=9x
        • -3/3 = -1
        • It turned out the expression 3x2 + 9x-1. It is not equal to the original expression.

   3. Write the original expression as equal to the product of gcd times the resulting expression. That is, enclose the resulting expression in brackets, and put the GCD out of brackets.

      • In our example: 9x 2 + 27x - 3 = 3(3x 2 + 9x - 1)

   4. Simplifying fractional expressions by taking the multiplier out of brackets. Why just take the multiplier out of brackets, as was done earlier? Then, to learn how to simplify complex expressions, such as fractional expressions. In this case, putting the factor out of the brackets can help get rid of the fraction (from the denominator).

      • For example, consider the fractional expression (9x 2 + 27x - 3)/3. Use parentheses to simplify this expression.
        • Factor out the factor 3 (as you did before): (3(3x 2 + 9x - 1))/3
        • Note that both the numerator and denominator now have the number 3. This can be reduced, and you get the expression: (3x 2 + 9x - 1) / 1
        • Since any fraction that has the number 1 in the denominator is just equal to the numerator, the original fractional expression is simplified to: 3x2 + 9x-1.

   Additional Simplification Techniques

4. Consider a simple example: √(90). The number 90 can be decomposed into the following factors: 9 and 10, and from 9, take the square root (3) and take 3 out from under the root.
   • √(90)
   • √(9×10)
   • √(9)×√(10)
   • 3×√(10)
   • 3√(10)

5. Simplifying expressions with powers. In some expressions, there are operations of multiplication or division of terms with a degree. In the case of multiplication of terms with one base, their degrees are added; in the case of dividing terms with the same base, their degrees are subtracted.

   • For example, consider the expression 6x 3 × 8x 4 + (x 17 / x 15). In the case of multiplication, add the exponents, and in the case of division, subtract them.
     • 6x 3 × 8x 4 + (x 17 / x 15)
     • (6 × 8)x 3 + 4 + (x 17 - 15)
     • 48x7+x2
   • The following is an explanation of the rule for multiplying and dividing terms with a degree.
     • Multiplying terms with powers is equivalent to multiplying terms by themselves. For example, since x 3 = x × x × x and x 5 = x × x × x × x × x, then x 3 × x 5 = (x × x × x) × (x × x × x × x × x), or x 8 .
     • Similarly, dividing terms with powers is equivalent to dividing terms by themselves. x 5 /x 3 \u003d (x × x × x × x × x) / (x × x × x). Since similar terms that are in both the numerator and the denominator can be reduced, the product of two "x", or x 2, remains in the numerator.

• Always be aware of the signs (plus or minus) in front of the terms of an expression, as many people have difficulty choosing the right sign.
• Ask for help if needed!
• Simplifying algebraic expressions is not easy, but if you get your hands on it, you can use this skill for a lifetime.

An algebraic expression in the record of which, along with the operations of addition, subtraction and multiplication, also uses division into literal expressions, is called a fractional algebraic expression. Such are, for example, the expressions

We call an algebraic fraction an algebraic expression that has the form of a quotient of division of two integer algebraic expressions (for example, monomials or polynomials). Such are, for example, the expressions

the third of the expressions).

Identity transformations of fractional algebraic expressions are for the most part intended to represent them as an algebraic fraction. To find a common denominator, the factorization of the denominators of fractions - terms is used in order to find their least common multiple. When reducing algebraic fractions, the strict identity of expressions can be violated: it is necessary to exclude the values ​​of quantities at which the factor by which the reduction is made vanishes.

Let us give examples of identical transformations of fractional algebraic expressions.

Example 1: Simplify an expression

All terms can be reduced to a common denominator (it is convenient to change the sign in the denominator of the last term and the sign in front of it):

Our expression is equal to one for all values ​​except these values, it is not defined and fraction reduction is illegal).

Example 2. Represent expression as an algebraic fraction

Solution. The expression can be taken as a common denominator. We find successively:

Exercises

1. Find the values ​​of algebraic expressions for the specified values ​​of the parameters:

2. Factorize.