Formulas of abbreviated multiplication. Detailed theory with examples

Mathematical expressions (formulas) abbreviated multiplication (square sums and differences, cube amount and differences, square differences, amount and difference of cubes) are extremely replaced in many areas exact Sciences. These 7 character recordings are not replaced by simplifying expressions, solving equations, with multiplication of polynomials, reducing fractions, solving integrals and many other things. So it will be very useful to figure out how they are obtained, for which they are needed, and most importantly, how to remember them and then apply. Then apply formulas of abbreviated multiplication In practice, the most difficult will see what is H.and what is u. Obviously, no restrictions for a. and b.no, which means it can be any numeric or letter expressions.

And so here they:

First x 2 - U 2. \u003d (x - y) (x + y) . To calculate square difference Two expressions need to multiply the difference between these expressions on their sums.

Second (x + y) 2 \u003d X 2. + 2h + in 2 . To find square amount Two expressions need to be added to the square of the first expression to add a double product of the first expression on the second plus the square of the second expression.

Third (x - y) 2 \u003d X 2. - 2h + in 2. To calculate square differencetwo expressions are needed from the square of the first expression to take away a double product of the first expression on the second plus the square of the second expression.

Fourth (x + y) 3 \u003d x 3. + 3x 2 y + 3h 2 + 3. To calculate cube amounttwo expressions need to be added to the Cuba of the first expression to add a tripled work of the square of the first expression on the second plus the tripled product of the first expression on the square plus cube of the second expression.

Fifth (x - y) 3 \u003d x 3. - 3x 2 y + 3h 2 - 3.. To calculate cube differencetwo expressions are necessary from the first expression cube to take the tripled work of the square of the first expression on the second plus the tripled product of the first expression on the second minus cube of the second expression.

Six x 3 + 3. \u003d (x + y) (x 2 - Hu + U 2) To calculate the amount of cubestwo expressions need to multiply the sums of the first and second expression on an incomplete square of the difference of these expressions.

Seventh x 3 - 3. \u003d (x - y) (x 2 + Hu + U 2) To make calculation cubic differencestwo expressions need to multiply the difference between the first and second expression on the incomplete square of the sum of these expressions.

It is not difficult to remember that all formulas are applied to the work of calculations and in the opposite direction (right to left).

About 4 thousand years ago on the existence of these patterns. They were widely used by residents of ancient Babylon and Egypt. But in those epochs, they expressed verbally or geometrically and during the calculations did not use the letters.

We will understand proof of Square Summa(a + b) 2 \u003d a 2 + 2ab + b 2.

First this mathematical pattern Proved an ancient Greek scientist Euclide, who worked in Alexandria in the III century BC, he used a geometric way to evof the formula, since scientists of ancient Ellala did not use the letters to designate numbers. They were universally used not "A 2", but "square on the segment A", not "AB", but "rectangle, concluded between segments A and B".

The linear function is called the function of the form y \u003d kx + b, where the X-independent variable, k and b-any numbers.
The graph of the linear function is straight.

1. To add schedule function, We need the coordinates of two points belonging to the graphics of the function. To find them, you need to take two values \u200b\u200bx, substitute them to the equation of the function, and to calculate the corresponding values \u200b\u200bof Y.

For example, to construct a graph of the function y \u003d x + 2, it is convenient to take x \u003d 0 and x \u003d 3, then the ordents of these points will be equal to y \u003d 2 and y \u003d 3. We obtain points A (0; 2) and in (3; 3). Connect them and obtain the graph of the function y \u003d x + 2:

2. In the formula y \u003d kx + b, the number K is called the coefficient of proportionality:
If k\u003e 0, then the function y \u003d kx + b increases
If K.
The coefficient B shows the displacement of the function schedule along the OY axis:
If b\u003e 0, then the function of the function y \u003d kx + b is obtained from the graph of the function \u003d kx shift to b units up along the Oy axis
If B.
Figure below shows the graphs of the functions Y \u003d 2X + 3; y \u003d ½ x + 3; y \u003d x + 3

Note that in all these functions the coefficient k above zero, and functions are increasing. Moreover, the greater the value K, the greater the angle of inclination direct to the positive direction of the OX axis.

In all functions B \u003d 3 - and we see that all the graphs cross the OY axis at the point (0; 3)

Now consider graphs of functions y \u003d -2x + 3; y \u003d - ½ x + 3; y \u003d -x + 3

This time in all functions of the K coefficient less than zero and functions decrease. The coefficient B \u003d 3, and graphics, as well as in the previous case, intersect the OY axis at the point (0; 3)

Consider graphs of functions y \u003d 2x + 3; y \u003d 2x; Y \u003d 2x-3

Now in all equations of functions, the coefficients k are equal to 2. and we got three parallel straight.

But B coefficients are different, and these graphs cross the OY axis at different points:
The graph of the function y \u003d 2x + 3 (B \u003d 3) crosses the OY axis at the point (0; 3)
The graph of the function y \u003d 2x (B \u003d 0) crosses the OY axis at the point (0; 0) - the beginning of the coordinates.
The graph of the function y \u003d 2x-3 (b \u003d -3) crosses the OY axis at the point (0; -3)

So, if we know the signs of the K and B coefficients, we can immediately imagine how the graph of the function y \u003d kx + b looks like.
If a k 0

If a k\u003e 0 and b\u003e 0 , then the graph of the function y \u003d kx + b is:

If a k\u003e 0 and b , then the graph of the function y \u003d kx + b is:

If a k, then the function of the function y \u003d kx + B has the form:

If a k \u003d 0. The function y \u003d kx + B turns into the Y \u003d B function and its graphic is:

The ordents of all points of the graph function y \u003d B are equal to b if b \u003d 0. , then the graph of the function y \u003d kx (direct proportionality) passes through the origin of the coordinate:

3. Separately, we note the graph of the equation x \u003d a. The graph of this equation is a straight line, parallel axis Oy all points of which have the abscissa x \u003d a.

For example, the graph of the equation x \u003d 3 looks like this:
Attention! Equation X \u003d A is not a function, so one value of the argument will meet different values Functions that does not correspond to the definition of the function.

4. The condition of parallelism of two straight lines:

Schedule of the function y \u003d k 1 x + b 1 parallel graphics of the function y \u003d k 2 x + b 2, if k 1 \u003d k 2

5. The condition of rebuilding the two straight lines:

The graph of the function y \u003d k 1 x + b 1 is rebuilt the graphics of the function y \u003d k 2 x + b 2, if k 1 * k 2 \u003d -1 or k 1 \u003d -1 / k 2

6. Points of intersection of the graph function y \u003d kx + b with axes of coordinates.

With OY axis. The abscissa of any point belonging to the OY axis is zero. Therefore, to find the intersection point with the OY axis, it is necessary to substitute zero in the equation. We get y \u003d b. That is, the intersection point with the Oy axis has coordinates (0; b).

With the axis oh: the ordinate of any point belonging to the axis Oh is equal to zero. Therefore, to find the intersection point with the axis oh, it is necessary to substitute zero in the equation of the function instead of y. We obtain 0 \u003d kx + b. Hence X \u003d -B / K. That is, the intersection point with the OX axis has coordinates (-b / k; 0):

Formulas of abbreviated multiplication. Workout.

Try to calculate the following expressions in this way:

Answers:

Or, if you know the squares of the main two-digit numbers, remember how much it will be? Remembered? . Excellent! Since we are erected into a square, then we must multiply on. Turns out that.

Remember that the formulas square sums and the square of the difference is valid not only for numerical expressions:

Calculate the following expressions:

Answers:

Formulas of abbreviated multiplication. Outcome.

Let's bring a small result and write the sum of the square of the sum and difference in one line:

Now it is practicing "collecting" the formula from the unfolded species in appearance. This skill will be needed in the future when transforming large expressions.

Suppose we have the following expression:

We know that the square of the amount (or difference) is square of one number square of another number and doubted work of these numbers.

In this task, it is easy to see the square of the same number - this is. Accordingly, one of the numbers included in the bracket is a square root of, that is,

Since in the second term there is, it means it is a double product of one and another, respectively:

Where is the second number in our bracket.

The second number included in the bracket is equal.

Check. should be equal. Indeed, it means that we found both numbers present in brackets: and. It remains to define a sign that stands between them. What do you think it will be behind the sign?

Right! Since we adjust Doubting work, then between numbers will stand a sign of addition. Now write a converted expression. Cope? You should get the following:

Note: the change of places of the components does not affect the result (no matter, addition or subtraction is between and).

It is absolutely optional that the terms in the transformed expression stood as written in the formula. Look at this expression :. Try to convert it yourself. Happened?

Practice - convert the following expressions:

Answers:Cope? Secure the topic. Choose from the expressions below that can be represented as a square amount or difference.

1. - Prove that it is equivalent.

1. - You can not imagine as a square; It would be possible to imagine if instead it was.

Square difference

Another formula of abbreviated multiplication is the difference of squares.

Square differences This is not a square of the difference!

The difference in the squares of the two numbers is equal to the amount of these numbers on their difference:

Check whether this formula is true. To do this, change the sum of the sum and difference in the removal of the Square formulas:

Thus, we just made sure that the formula is really true. This formula also simplifies complex computational actions. Let us give an example:

It is necessary to calculate :. Of course, we can build a square, then build a square and subtract one of the other, but the formula simplifies us the task:

Happened? Complete Results:

As well as the sum of the sum (difference), the square difference formula can be applied not only with numbers:

The ability to spread the difference in squares will help us convert complex mathematical expressions.

Pay attention:

Since, when decomposing the square of the difference of the right expression, we will get

Be careful and see which concrete term is being built into a square! To secure the theme converting the following expressions:

Record? Compare the obtained expressions:

Now, when you learned the square of the amount and square of the difference, as well as the difference of squares, try to solve examples on the combination of these three formulas.

Converting elementary expressions (square sum, square square, square difference)

Suppose we are given an example

It is necessary to simplify this expression. Look carefully, what do you see in the numerator? That's right, the numerator is a full square:

Simplifying expression, remember that a hint, which way to move in simplification is in the denominator (or in a numerator). In our case, when the denominator is decomposed, and it is impossible to do anything else, it can be understood that the numerator will be either the square of the amount, or the square of the difference. Since we add, it becomes clear that the numerator is the square of the amount.

Try to independently convert the following expressions:

Happened? Compare answers and move on!

Cube amount and Cube difference

Formulas Cube amount and Cube difference are derived similarly as square amount and square difference: Disclosure of brackets with multiplies members to each other.

If the square of the amount and the square of the difference remember very easily, then the question arises "How to remember Cubes?"

Look carefully on two described formulas in comparison with the construction of similar members in the square:

What kind of pattern do you see?

1. When erected in square we have square The first number I. square second; When erecting in the cube - there cubic one number I. cubic another number.

2. When erected in square, we have doubts the product of numbers (the number of 1 degree, which is one degree less than the one in which the expression is erected); When erecting B. cubic - tripled The work in which one of the numbers is erected into a square (which is also 1 degree less than the degree in which the expression is erected).

3. When it is erected in the square, the sign in brackets in the disclosed expression is reflected in addition (or subtracting) a double product - if addition in brackets, then add, if subtraction - take away; When erected into the cube, the rule is: if we have a cube amount, then all the signs "+", and if a cube difference, then signs alternate: "" - "" - "" - "".

All of the above, except for the dependence of degrees when multiplying members, is shown in the figure.

Practice? Cutting brackets in the following expressions:

Compare the obtained expressions:

Difference and amount of cubes

Consider the last pair of formulas the difference and amount of cubes.

As we remember, in the difference of squares we have multiplying the difference and the amount of these numbers one to another. In the difference of cubes and in the amount of cubes there are also two brackets:

1 bracket - the difference (or amount) of numbers in the first degree (depending on the difference or amount of cubes we reveal);

2 bracket - incomplete square (Look: if we were subtracted (or added) a double product of numbers, there would be a square), a sign when multiplying the numbers is the opposite sign of the initial expression.

To secure the topic, solve several examples:

Compare the obtained expressions:

Workout

Answers:

Let's summarize:

There are 7 formulas of abbreviated multiplication:

ADVANCED LEVEL

The formulas of abbreviated multiplication are formulas, knowing which one can avoid performing some standard actions when simplifying expressions or decomposition of polynomials to multipliers. Formulas of the abbreviated multiplication you need to know by heart!

1. Square amount Two expressions equal to square The first expression plus a twisted work of the first expression on the second plus the square of the second expression:
2. Square difference Two expressions is equal to the square of the first expression minus a twice product of the first expression on the second plus the square of the second expression:
3. Square difference Two expressions are equal to the product of these expressions and their sum:
4. Cube amount Two expressions are equal to Cuba of the first expression plus the tripled product of the square of the first expression on the second plus the tripled product of the first expression on the square of the second plus cube of the second expression:
5. Cube difference Two expressions are equal to Cuba of the first expression minus the tripled work of the square of the first expression on the second plus the tripled product of the first expression on the square of the second minus cube of the second expression:
6. The amount of cubes Two expressions are equal to the amount of the sum of the first and second expression on an incomplete square of the difference of these expressions:
7. Cubic differences Two expressions are equal to the product of the first and second expression on an incomplete square of the sum of these expressions:

Now we will prove all these formulas.

Formulas of abbreviated multiplication. Evidence.

1. .
Evaluate the expression to the square - it means to multiply it by itself:
.

We will reveal the brackets and give similar:

2. .
We do the same: multiply the difference for yourself, we reveal the brackets and give these things:
.

3. .
Take the expression on the right side and reveal the brackets:
.

4. .
The number in Cuba can be represented as a multiplied number to its square:

Similarly:

In the difference of cubes, signs alternate.

6. .

.

7. .
We will open brackets in the right part:
.

Application of formulas of abbreviated multiplication when solving examples

Example 1:

Find the value of expressions:

Decision:

1. We use the formula Square amount :.
2. Imagine this number in the form of a difference and use the formula of the square of the difference :.

Example 2:

Find the value of the expression :.

Decision:

Using the formula of the size of the squares of two expressions, we get:

Example 3:

Simplify the expression:

Solution in two ways:

We use the formulas square sums and square differences:

II way.

We use the formula for the difference in the squares of two expressions:

Now your word ...

I told everything that I know about the formulas of abbreviated multiplication.

Tell me now you will use them? If not, why?

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Write in the comments. We read all comments and answer everything.

And good luck on the exams!

In the previous lesson, we dealt with the decomposition of multipliers. Two ways were mastered: making a common factor for brackets and grouping. In this lesson - the next powerful way: formulas of abbreviated multiplication. In a brief record - FSU.

The formulas of the abbreviated multiplication (the square of the sum and difference, the cube of the amount and difference, the difference of squares, the sum and the difference of cubes) are extremely necessary in all sections of mathematics. They are used in simplifying expressions, solving equations, multiplication of polynomials, reduction of fractions, solving integrals, etc. etc. In short, there is every reason to deal with them. To understand how they are taken, why are they needed, how to remember them and how to apply.

We understand?)

Where do abbreviated multiplication formulas come from?

Equality 6 and 7 are not written very familiar. As if on the contrary. This is specially.) Any equality works both from left to right and right to left. In such a record, it is clear where the FSU comes from.

They are taken from multiplication.) For example:

(A + B) 2 \u003d (A + B) (A + B) \u003d A 2 + AB + BA + B 2 \u003d A 2 + 2AB + B 2

That's all, no scientific tricks. Just change the brackets and give these. So it turns out all formulas of abbreviated multiplication. Abbreviated Multiplication is because in the formulas themselves there is no multiplication of brackets and bringing similar. Reduced.) Immediately given the result.

FSU need to know by heart. Without the first three, you can not dream about the troika, without the rest - about the fourth with a five.)

Why do the formulas of abbreviated multiplication need?

There are two reasons, learn, even to get these formulas. The first - the finished answer on the machine sharply reduces the number of errors. But this is not the most main reason. But the second ...

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