Addition and subtraction of monomials. Video lesson “Arithmetic operations with monomials

Slide 3

Slide 4

Stage 1: “Repetition is the mother of learning” Decipher the word: ALGEBRA from the Arabic word “Al” - jebra” (translated as “restoration.”)

Slide 5

Slide 6

1. A monomial is the sum of numerical and alphabetic factors. 2. All numbers, any variables, powers of variables are also considered monomials. 3. The literal factor of a monomial written in standard form is called the coefficient of the monomial. 4. Algebraic expression, which is the product of numbers and variables raised to powers with natural indicator, is called a monomial

Slide 7

5. The sum of the exponents of all letters included in the monomial is called the degree of the monomial. 6. Identical or differing from each other only in coefficients are called similar terms. 7. Two monomials consisting of the same variables are called similar monomials. 8. As a result of adding monomials, a monomial is obtained.

Slide 8

9. A monomial in which all numerical factors are multiplied and their product is placed in first place, all available powers with the same letter base are multiplied, and all powers with a different letter base are multiplied is called a monomial of standard form. 10. To open brackets preceded by a “+” sign, the brackets must be omitted, preserving the sign of each term that was enclosed in brackets. 11. When we open brackets preceded by a “-” sign, we omit the brackets and the signs of the members that were enclosed in brackets are reversed.

Slide 9

Slide 10

Find the error:

Slide 11

From the written monomials, select similar ones and find their sum:

Slide 12

A D U G S I

Slide 13

The first stage is drawing up mathematical model. (SMM) Let the whole distance be x km, then on the first day we walked On the second day we walked

Slide 14

Since there are 25 km left on the third day, we get a mathematical model: The second stage is working with the compiled model. RMM

Slide 15

2. RMM Stage 3: Answer to the question of the problem: (OVZ) We took the length of the path as x, which means it is equal to 55 km. Answer: the length of the path is 55 km.

Slide 16

A Z D U G S I

Slide 17

“A book is a book, but move your brains” No. 292 No. 293

In this lesson we will remember what a monomial is, the standard form of a monomial, and give a definition of similar monomials. Let's learn to distinguish similar monomials from dissimilar ones. Let us formulate the rules for adding and subtracting similar monomials. Let's learn to solve typical tasks using addition and subtraction.

Subject:Monomials. Arithmetic operations on monomials

Lesson:Adding and subtracting monomials

Let's remember what is called a monomial, and what operations can be done with monomials. A monomial is the product of numbers and powers. Let's look at two examples:

Both expressions are monomials and before proceeding with addition or subtraction, it is necessary to bring them to standard form:

Recall that to reduce a monomial to standard form, you must first obtain numerical coefficient, multiplying all numerical factors, and then multiplying the corresponding powers.

Let's find out whether it is possible to add our two monomials - no, it is not possible, because you can add only those monomials that have the same letter part, that is, only similar monomials. That is, we must learn to distinguish between similar and non-similar monomials.

Let's look at examples of similar monomials:

Monomials and are similar because they have the same letter part -

Another example. Let's write a monomial and a monomial. We can assign absolutely any numerical coefficient to the second monomial and get a monomial similar to the first. Let us choose, for example, a coefficient and get two similar monomials: and

Consider the following example. First monomial, its coefficient equal to one. Let us now write down its letter part and add to it an arbitrary numerical coefficient, for example, . We have two similar monomials: and .

Let's do it conclusion: Similar monomials have the same letter part, and such monomials can be added and subtracted.

Now we give examples of non-similar monomials:

AND ; these monomials have different letter parts, the variable a in them is represented in different degrees, so the monomials are not similar

Another example: the monomials and are also not similar; their letter parts differ in powers of the variable a.

Let's consider the third pair of monomials: and are also not similar.

Now let’s look at the addition of similar monomials; to do this, let’s do an example:

Add two monomials:

It is obvious that these monomials are similar, since it is easy to notice that their letter parts are the same, but mathematically the similarity of the monomials can be proven by replacing the letter part with another letter, and if for both monomials this letter turns out to be the same, then the monomials are similar. Moving on to an example, let’s replace the first monomial with ? Then in the second monomial we replace the same letter part with

Adding these two expressions, we get . Now let's return to the original variables - replace the variable t in the answer with , we get the final answer:

Now let's formulate rule for adding monomials:

In order to obtain the sum of similar monomials, it is necessary to add their coefficients, and add the letter part the same as for the original terms.

Let's look at examples:

2)

Comment on example No. 1: first we write the sum of the coefficients of the monomials into the result, that is, then we rewrite the literal part without changes, that is

Comment on example No. 2: similar to the first example, we first write down the sum of the coefficients, that is, then rewrite the letter part without changes - .

Let's move on to rule for subtracting monomials. Consider examples:

The rule for subtracting such monomials is similar to the rule for addition: we rewrite the letter part without changes, and subtract the coefficients, and subtract them in the correct order. For our example:

Let's do it conclusion: You can add and subtract any monomials, but only similar ones; to do this you need to add or subtract their coefficients, rewriting the letter part in its original form. Non-similar monomials cannot be added or subtracted.

Now, knowing the algorithm for adding and subtracting similar monomials, we can solve some typical problems.

Simplification tasks:

Simplify the expression:

The first monomial is written in a standard form, it cannot be simplified any more, the second and third are not in a standard form, which means that the first action when simplifying expressions with monomials is to reduce the monomials that can be reduced to it to a standard form.

So, let’s bring the second and then the third monomials to standard form:

Let's rewrite the original expression taking into account the transformations performed:

We see the same letter part for all three monomials, which means they are similar, that is, we have the right to add and subtract them. According to the rule, we will fulfill necessary actions with coefficients, and rewrite the literal part without changes:

Exists inverse problem . A monomial is given. Represent a monomial as a sum of monomials.

All monomials, in the form of the sum of which we present the given one, will have the same letter part, which is also the same with the given monomial - . Let's imagine our monomial, for example, as a sum of two terms. To do this, let's imagine the coefficient as a sum.

Let’s continue our acquaintance with monomials with the material in the article below: let’s look at the implementation basic actions with monomials such as addition and subtraction. Let's consider in what cases these actions must be performed and what they will give in the end; Let's formulate the rule of addition and subtraction and apply it to solve standard problems.

Result of adding and subtracting monomials

We will study addition and subtraction of monomials based on operations with polynomials, since, in general, the result of addition or subtraction of monomials is a polynomial, and only in special situations is a monomial.

In other words, addition and subtraction on a set of monomials can be introduced only with restrictions. Let's clarify what this means by drawing an analogy with subtracting natural numbers. On the set of natural numbers, the action of subtraction is also considered with a limitation: in order for the result to become a natural number, the subtraction must be performed only according to the scheme: from a larger natural number less.

It's another matter if we're talking about about the set of integers, including natural numbers: here subtraction is carried out without restrictions.

The same can be applied when it comes to adding or subtracting two monomials. In order to ultimately obtain a monomial, addition or subtraction on a set of monomials can be carried out with a restriction: the original added or subtracted monomials must be similar terms (then they are called similar monomials), or one of them must be zero. In other cases, the result of actions is no longer a monomial.

But on the set of polynomials, which contains all monomials, addition and subtraction of monomials is studied as a special case of addition and subtraction of polynomials. In this case, actions are considered without the above restrictions, since the result of their execution is a polynomial (or a monomial as special case polynomial).

Rule for adding and subtracting monomials

Let us formulate the rule for adding and subtracting monomials in the form of a sequence of actions:

Definition 1

To carry out the action of adding or subtracting two monomials you must:

• write down the sum or difference of monomials depending on the task: monomials must be enclosed in brackets, placing a plus or minus sign between them, respectively;
• if monomials in brackets are present in non-standard form, bring them to a standard form;
• open parentheses;
• Give similar terms, if any, and eliminate terms that are equal to zero.

Now let’s apply the stated rule to solve problems.

Examples of adding and subtracting monomials

Monomials given 8 x And − 3 x. It is necessary to perform their addition and subtraction.

Solution

1. Let's perform the addition action. Let's write the sum by enclosing the original monomials in parentheses and putting a plus sign between them: (8 x) + (− 3 x). The monomials in brackets have a standard form, which means that the second step of the rule algorithm can be skipped. The next step is to open the brackets: 8 x − 3 x, and then we present similar terms: 8 x − 3 x = (8 − 3) x = 5 x.

Let us briefly write the solution as follows: (8 x) + (− 3 x) = 8 x − 3 x = 5 x.

1. Let's perform the subtraction operation in the same way: (8 x) − (− 3 x) = 8 x + 3 x = 11 x.

Answer: (8 x) + (− 3 x) = 5 x And (8 x) − (− 3 x) = 11 x.

Let's consider an example where one of the monomials is zero.

Example 2

It is necessary to find the difference between the monomial - 5 · x 3 · 2 3 · 0 · x · z 2 and the monomial x · 2 3 · y 5 · z · - 3 8 · x · y.

Solution

We act according to the algorithm according to the rule. Let's write the difference: - 5 · x 3 · 2 3 · 0 · x · z 2 - x · 2 3 · y 5 · z · - 3 8 · x · y. We bring the monomials enclosed in brackets to standard form and then we get: 0 - - 1 4 · x 2 · y 6 · z. Let's open the brackets, which will give us the following form of expression: 0 + 1 4 · x 2 · y 6 · z, it, due to the property of adding zero, will be identically equal to 1 4 · x 2 · y 6 · z.

Thus, short note the solution will be like this:

5 x 3 2 3 0 x z 2 - x 2 3 y 5 z - 3 8 x y = = 0 - - 1 4 x 2 y 6 z = 1 4 · x 2 · y 6 · z

Answer:- 5 x 3 2 3 0 x z 2 - x 2 3 y 5 z - 3 8 x y = 1 4 x 2 y 6 z

The considered examples yielded monomials as a result of addition and subtraction. However, as already mentioned, in general case the result of addition and subtraction is a polynomial.

Example 3

Monomials given − 9 x z 3 And − 13 x y z. It is necessary to find their sum.

Solution

We write down the amount: (− 9 x z 3) + (− 13 x y z). Monomials have a standard form, so we expand the brackets: (− 9 · x · z 3) + (− 13 · x · y · z) = − 9 · x · z 3 − 13 · x · y · z . There are no similar terms in the resulting expression, we have nothing to give, which means the resulting expression will be the result of the calculation: − 9 · x · z 3 − 13 · x · y · z.

Answer: (− 9 x z 3) + (− 13 x y z) = − 9 x z 3 − 13 x y z.

The same scheme applies to the addition or subtraction of three or more monomials.

Example 4

An example needs to be solved: 0 , 2 · a 3 · b 2 + 7 · a 3 · b 2 − 3 · a 3 · b 2 − 2 , 7 · a 3 · b 2.

Solution

All given monomials have a standard form and are similar. Let's give similar members by performing addition and subtraction numerical coefficients, and leaving the letter part as original: 0 , 2 · a 3 · b 2 + 7 · a 3 · b 2 − 3 · a 3 · b 2 − 2 , 7 · a 3 · b 2 = = (0 , 2 + 7 − 3 − 2 , 7) a 3 b 2 = 1, 5 a 3 b 2

Answer: 0, 2 · a 3 · b 2 + 7 · a 3 · b 2 − 3 · a 3 · b 2 − 2, 7 · a 3 · b 2 = 1, 5 · a 3 · b 2.

Example 5

The monomials are given: 5, − 3 a, 15 a, − 0, 5 x z 4, − 12 a, − 2 and 0.5 x z 4. It is necessary to find their sum.

Solution

Let's write down the amount: (5) + (− 3 a) + (15 a) + (− 0.5 x z 4) + (− 12 a) + (− 2) + (0.5 x z 4 ). As a result of expanding the brackets, we get: 5 − 3 a + 15 a − 0, 5 x z 4 − 12 a − 2 + 0, 5 x z 4. Let's group similar terms: (5 − 2) + (− 3 a + 15 a − 12 a) + (− 0.5 x z 4 + 0.5 x z 4) and let's list them: 3 + 0 + 0 = 3

Answer: (5) + (− 3 a) + (15 a) + (− 0.5 x z 4) + (− 12 a) + (− 2) + (0.5 x z 4 ) = 3.

If you notice an error in the text, please highlight it and press Ctrl+Enter

Adding monomials or subtracting one monomial from another is possible only if the monomials are similar. If the monomials are not similar, in this case the addition of monomials can be written as a sum, and the subtraction as a difference.

Similar monomials

Similar monomials- monomials, which consist of the same letters, but may have different or the same coefficients (numerical factors). Identical letters in similar monomials must have same indicators degrees. If the degrees of the same letter in different monomials do not coincide, then such monomials cannot be called similar:

5ab 2 and -7 ab 2 - similar monomials

5a 2 b and 5 ab - not similar monomials

Please note that the sequence of letters in similar monomials may not be the same. Also, monomials can be represented in the form of an expression that can be simplified, therefore, before you begin to determine whether these monomials are similar or not, it is worth bringing the monomials to a standard form. For example, let's take two monomials:

5abb and -7 b 2 a

Both monomials are in non-standard form, so it will not be easy to determine whether they are similar. To find out, let’s reduce the monomials to standard form:

5ab 2 and -7 ab 2

Now it is immediately clear that these monomials are similar.

Two similar monomials that differ only in sign are called opposite. For example:

5a 2 bc and -5 a 2 bc- opposite monomials.

Reducing similar monomials is a simplification of an expression containing similar monomials by adding them. The addition of similar monomials is carried out according to the rules for reducing similar terms.

Addition of monomials

To add monomials you need:

1. Create a sum by writing all the terms one after the other
2. To bring similar terms, for this you need:

Example 1. Add monomials 12 ab, -4a 2 b and -5 ab.

Solution: Let's make the sum of the monomials:

12ab + (-4a 2 b) + (-5ab)

12ab - 4a 2 b - 5ab

Now we need to determine whether there are similar monomials among the terms and, if there are any, make a reduction:

12ab - 4a 2 b - 5ab = (12 + (-5))ab - 4a 2 b = 7ab - 4a 2 b

Example 2. Add monomials 5 a 2 bc and -5 a 2 bc.

Solution: Let's make the sum of the monomials:

5a 2 bc + (-5a 2 bc)

Let's expand the brackets:

5a 2 bc - 5a 2 bc

These two monomials are opposites, that is, they differ only in sign. This means that if we add their numerical factors, we get zero:

5a 2 bc - 5a 2 bc = (5 - 5)a 2 bc = 0a 2 bc = 0

Hence, when adding opposite monomials the result is zero.

General rule addition of monomials:

To add several monomials, you should write down all the terms one after the other, preserving their signs, put the negative monomials in parentheses, and make a reduction similar terms(similar monomials).

Subtracting monomials

To subtract monomials you need to:

1. Compose the difference by writing all the monomials one after another, separating them with a - (minus) sign.
2. Bring all monomials to standard form
3. Expand parentheses if they are in the expression
4. Make a reduction of similar monomials, that is:
   1. add their numerical factors
   2. After the resulting coefficient, add letter factors without changes

Example. Find the difference of monomials 8 ab 2 , -5a 2 b And - ab 2 .

Solution: Let's make up the difference of monomials:

8ab 2 - (-5a 2 b) - (-ab 2)

All monomials are in standard form. So you can start opening the brackets. See the rules for opening brackets.

8ab 2 + 5a 2 b + ab 2

Now we need to determine whether there are similar ones among the monomials and, if they are, make a reduction:

8ab 2 + 5a 2 b + ab 2 = (8 + 1)ab 2 + 5a 2 b = 9ab 2 + 5a 2 b

The general rule for subtracting monomials:

To subtract one monomial from another, add the subtrahend monomial with to the minuend opposite sign and make a reduction of similar monomials.