If one of the factors is zero, then the product is zero. Twice two is four, or Multiplication and division The product is equal to zero factors

Along with addition, important operations are multiplication and division. Let us recall at least the problem of determining how many times Masha has more apples than Sasha, or finding the number of parts produced per year, if the number of parts produced per day is known.

Multiplication Is one of four basic arithmetic operations, during which one number is multiplied by another. In other words, the record 5 · 3 = 15 means that the number 5 was folded 3 times, i.e. 5 · 3 = 5 + 5 + 5 = 15.

The multiplication is regulated by the system rules.

1. Product of two negative numbers equals positive number... To find the modulus of the product, you need to multiply the moduli of these numbers.

(- 6) ( - 6) = 36; (- 17.5) ( - 17,4) = 304,5

2. The product of two numbers with different signs equals a negative number. To find the modulus of the product, you need to multiply the moduli of these numbers.

(- 5) 6 = - thirty; 0.7 ( - 8) = - 21

3. If one of the factors is zero, then the product is zero. The converse is also true: the product is zero only if one of the factors is zero.

2.73 * 0 = 0; ( - 345.78) 0 = 0

Based on the above material, we will try to solve the equation 4 ∙ (x – 5) = 0.

1. Let's open the brackets and get 4x - 20 = 0.

2. Move (-20) to the right side (do not forget to change the sign to the opposite) and
we get 4x = 20.

3. Find x by canceling both sides of the equation by 4.

4. Total: x = 5.

But knowing rule # 3, we can solve our equation much faster.

1. Our equation is 0, and according to rule number 3, the product is 0 if one of the factors is 0.

2. We have two factors: 4 and (x - 5). 4 is not equal to 0, so x - 5 = 0.

3. We solve the resulting simple equation: x - 5 = 0. Hence, x = 5.

Multiplication relies on two laws - transpositional and combination laws.

Traveling law: for any numbers a and b equality is true ab = ba:

(- 6) 1.2 = 1.2 ( - 6), i.e. = - 7,2.

Combination law: for any numbers a, b and c equality is true (ab) c = a (bc).

(- 3) ( - 5) 2 = ( - 3) (2 ( - 5)) = (- 3) ( - 10) = 30.

The inverse of multiplication is division... If the components of the multiplication are called multipliers, then the division of the number that is divisible is called divisible, the number by which we divide - divider and the result is private.

12: 3 = 4, where 12 is the dividend, 3 is the divisor, 4 is the quotient.

Division, similar to multiplication, is adjustable the rules.

1. The quotient of two negative numbers is a positive number. To find the modulus of the quotient, you need to divide the modulus of the dividend by the modulus of the divisor.

- 12: (- 3) = 4

2. The quotient of two numbers with different signs is a negative number. To find the modulus of the quotient, you need to divide the modulus of the dividend by the modulus of the divisor.

- 12: 3 = - 4; 12: (- 3) = - 4.

3. Dividing zero by any nonzero number results in zero. You cannot divide by zero.

0: 23 = 0; 23: 0 = XXXX

Based on the division rules, let's try to solve an example - 4 x ( - 5) – (- 30) : 6 = ?

1. Perform multiplication: -4 x (-5) = 20. So, our example will take the form 20 - (-30): 6 =?

2. Perform division (-30): 6 = -5. This means that our example will take the form 20 - (-5) =?.

3. Subtract 20 - (-5) = 20 + 5 = 25.

So our the answer is 25.

Knowledge of multiplication and division, along with addition and subtraction, allows us to solve various equations and problems, as well as perfectly navigate the world of numbers and operations around us.

Let's fix the material by deciding equation 3 ∙ (4x – 8) = 3x – 6.

1. Let's open the brackets 3 ∙ (4x - 8) and get 12x - 24. Our equation has become 12x - 24 = 3x - 6.

2. Here are similar ones. To do this, move all components from x to the left, and all numbers to the right.
We get 12x - 24 = 3x - 6 → 12x - 3x = -6 + 24 → 9x = 18.

Do NOT forget to change the signs to the opposite when transferring a component from one side of the equation to another.

3. We solve the resulting equation 9x = 18, whence x = 18: 9 = 2. So, our answer is 2.

4. To make sure that our decision is correct, we will check:

3 ∙ (4x - 8) = 3x - 6

3 (4 ∙ 2 - 8) = 3 ∙ 2 - 6

3 ∙ (8 – 8) = 6 – 6

0 = 0, which means our answer is correct.

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What is it in outward appearance equations determine whether this equation will be incomplete quadratic equation? But as solve incomplete quadratic equations?

How to find out "by sight" an incomplete quadratic equation

Left part of the equation is square trinomial, a right — number 0. Such equations are called complete quadratic equations.

Have complete quadratic equation all odds, and not equal 0. There are special formulas for their solution, which we will get acquainted with later.

Most simple for the solution are incomplete quadratic equations. These are quadratic equations in which some coefficients are zero.

Coefficient by definition cannot be zero, otherwise the equation will not be quadratic. We talked about this. It means that it turns out that to turn to zero may only odds or.

Depending on this, there is three types of incomplete quadratic equations.

1) , where ;
2) , where ;
3) , where .

So, if we see a quadratic equation, on the left side of which instead of three members present two members or one member, then such an equation will be incomplete quadratic equation.

Determining an Incomplete Quadratic Equation

Incomplete quadratic equation is called a quadratic equation in which at least one of the coefficients or is zero.

This definition contains a very important the phrase " at least one from the coefficients ... is zero". It means that one or more coefficients can be equal zero.

Based on this, it is possible three options: or one the coefficient is zero, or another the coefficient is zero, or both coefficients are simultaneously equal to zero. This is how we get three types of incomplete quadratic equation.

Incomplete quadratic equations are the following equations:
1)
2)
3)

Equation solution

Outline solution plan of this equation. Left part of the equation can be easily factor, since the terms on the left side of the equation have common factor, it can be taken out of the parenthesis. Then the product of two factors is obtained on the left, and zero on the right.

And then the rule “the product is equal to zero if and only if at least one of the factors is equal to zero, and the other makes sense” will work. Everything is very simple!

So, solution plan.
1) Factor the left side.
2) We use the rule "the product is equal to zero ..."

Equations of this type I call "a gift of fate"... These are equations for which the right side is zero, a left part can be expanded by factors.

Solving the equation according to plan.

1) Let's expand the left side of the equation by factors, for this we take out the common factor, we get the following equation.

2) In the equation we see that left costs work, a right zero.

Real a gift of fate! Here, of course, we will use the rule "the product is equal to zero if and only if at least one of the factors is equal to zero, and the other makes sense".

When translating this rule into the language of mathematics, we get two equations or.

We see that the equation disintegrated for two more simple equations, the first of which has already been solved ().

Let's solve the second the equation . Move the unknown terms to the left, and the known ones to the right. The unknown member is already on the left, we will leave it there. And we will move the known term to the right with the opposite sign. Let's get the equation.

We have found, but we need to find. To get rid of the factor, you need to divide both sides of the equation by.

"Parallelism of two lines" - Prove that AB || CD. C - secant for a and b. BC is the bisector of the angle ABD. Will there be m || n? Examples of concurrency in real life... Are the lines parallel? Name the pairs: - criss-cross lying corners; - corresponding angles; - one-sided corners; The first sign of parallelism of straight lines. Prove that AC || BD.

"Two frosts" - Well, I think, wait with me now. Two frosts. And in the evening we met again in an open field. Frost shook his head - Blue nose and said: - Eh, you are young, brother, and stupid. Let him, how he dresses, let him know what Frost is - Red nose. Live with mine, so you will find out that an ax warms better than a fur coat. Well, I think we'll get to the place, then I'll grab you.

"Linear Equation in Two Variables" - Definition: Linear Equation in Two Variables. Algorithm for proving that a given pair of numbers is a solution to an equation: Give examples. -What equation in two variables is called linear? -What is called an equation in two variables? Equality containing two variables is called an equation in two variables.

"Interference of two waves" - Interference. Cause? Thomas Jung's experience. Interference of mechanical waves on water. Wavelength. Light interference. A stable interference pattern is observed if the superimposed waves are coherent. Radio telescope-interferometer located in New Mexico, USA. Application of interference. Interference of mechanical sound waves.

"Sign of perpendicularity of two planes" - Exercise 6. Perpendicularity of planes. Answer: Yes. Is there a triangular pyramid in which three faces are pairwise perpendicular? Exercise 1. Find the angles ADB and ACB. Answer: 90 °, 60 °. Exercise 10. Exercise 3. Exercise 7. Exercise 9. Is it true that two planes perpendicular to the third are parallel?

“Inequalities in two variables” - The geometric model of solutions to inequalities is the middle region. Lesson goal: Solutions to inequalities with two variables. 1.Plot the equation f (x, y) = 0. To solve inequalities with two variables, a graphical method is used. The circles have divided the plane into three areas. An inequality with two variables most often has an infinite number of solutions.

If one and two factors are equal to 1, then the product is equal to the other factor.

III. Working on new material.

Pupils can explain the multiplication technique for cases when there are zeros in the middle of writing a multi-digit number: for example, the teacher suggests calculating the product of 907 and 3. Pupils write down the solution in a column, arguing: “I write the number 3 under ones.

I multiply the number of ones by 3: three times seven - 21, that's 2 dess. and 1 unit; I write 1 under units, and 2 dess. remember. I multiply tens: 0 multiplied by 3, it turns out 0, and 2 more, it turns out 2 tens, I write 2 under tens. I multiply hundreds: 9 times 3, I get 27, I write 27. I read the answer: 2 721. "

To consolidate the material, the students solve the examples from task 361 with a detailed explanation. If the teacher sees that the children have dealt with the new material well, then he can offer a short commentary.

Teacher. We will explain the solution briefly, name only the number of units of each digit of the first multiplier that you multiply, and the result, without naming which digit of these units. Let's multiply 4 019 by 7. I explain: I will multiply 9 by 7, I will get 63, I write 3, I memorize 6. I multiply by 7, it turns out 7, and even 6 is 13, I write 3, I remember 1. Zero multiplied by 7, it turns out zero, and moreover 1, I get 1, I write 1. 4 I multiply by 7, I get 28, I write 28. I read the answer: 28 133.

F i z k u l t m and n u t k a

IV. Work on the material covered.

1. Solving problems.

Students solve Problem 363 with comments. After reading the task, a short condition is written.

The teacher can ask students to solve a problem in two ways.

Answer: 7,245 quintals of grain were harvested in total.

Children solve Problem 364 on their own (with subsequent verification).

1) 42 10 = 420 (q) - wheat

2) 420: 3 = 140 (q) - barley

3) 420 - 140 = 280 (q)

Answer: 280 quintals of wheat more.

2. Solution of examples.

The children carry out task 365 independently: they write down expressions and find their meanings.

V. Lesson summary.

Teacher. Guys, what have you learned in the lesson?

Children. We got acquainted with a new multiplication technique.

Teacher. What was repeated in the lesson?

Children. We solved problems, composed expressions and found their meanings.

Homework: tasks 362, 368; notebook number 1, p. 52, no. 5-8.

Level 58
Multiplication of numbers written
ends with zeros

Goals: to acquaint with the technique of multiplication by a single-digit number of multi-digit numbers ending in one or more zeros; to consolidate the ability to solve problems, examples for division with a remainder; repeat the table of time units.