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Abbreviated multiplication formulas. Detailed theory with examples

08.07.2023

Mathematical expressions (formulas) abbreviated multiplication(the square of the sum and difference, the cube of the sum and difference, the difference of squares, the sum and difference of cubes) are extremely irreplaceable in many areas of the exact sciences. These 7 character entries are irreplaceable when simplifying expressions, solving equations, multiplying polynomials, reducing fractions, solving integrals and much more. So it will be very useful to figure out how they are obtained, what they are for, and most importantly, how to remember them and then apply them. Then applying abbreviated multiplication formulas in practice, the most difficult thing will be to see what is X and what have. Obviously there are no restrictions on a And b no, which means it can be any numeric or literal expression.

And so here they are:

First x 2 - at 2 = (x - y) (x + y).To calculate difference of squares two expressions, it is necessary to multiply the differences of these expressions by their sums.

Second (x + y) 2 = x 2 + 2xy + y 2. To find sum squared two expressions, you need to add to the square of the first expression twice the product of the first expression by the second plus the square of the second expression.

Third (x - y) 2 = x 2 - 2xy + y 2. To calculate difference squared two expressions, you need to subtract from the square of the first expression twice the product of the first expression by the second plus the square of the second expression.

Fourth (x + y) 3 = x 3 + 3x 2 y + 3x 2 + at 3. To calculate sum cube two expressions, you need to add to the cube of the first expression three times the product of the square of the first expression and the second, plus three times the product of the first expression and the square of the second, plus the cube of the second expression.

Fifth (x - y) 3 = x 3 - 3x 2 y + 3x 2 - at 3. To calculate difference cube two expressions, it is necessary to subtract from the cube of the first expression the triple product of the square of the first expression by the second plus the triple product of the first expression by the square of the second minus the cube of the second expression.

sixth x 3 + y 3 = (x + y) (x 2 - xy + y 2) To calculate sum of cubes two expressions, you need to multiply the sums of the first and second expressions by the incomplete square of the difference of these expressions.

seventh x 3 - at 3 \u003d (x - y) (x 2 + xy + y 2) To make a calculation cube differences two expressions, it is necessary to multiply the difference of the first and second expressions by the incomplete square of the sum of these expressions.

It is not difficult to remember that all formulas are used to make calculations in the opposite direction (from right to left).

The existence of these regularities was known about 4 thousand years ago. They were widely used by the inhabitants of ancient Babylon and Egypt. But in those eras they were expressed verbally or geometrically and did not use letters in calculations.

Let's analyze sum square proof(a + b) 2 = a 2 + 2ab + b 2 .

This mathematical regularity proved the ancient Greek scientist Euclid, who worked in Alexandria in the 3rd century BC, he used the geometric method of proving the formula for this, since the scientists of ancient Hellas did not use letters to denote numbers. They everywhere used not “a 2”, but “square on segment a”, not “ab”, but “rectangle enclosed between segments a and b”.

Formulas or rules of reduced multiplication are used in arithmetic, and more specifically in algebra, for a faster process of calculating large algebraic expressions. The formulas themselves are derived from the existing rules in algebra for the multiplication of several polynomials.

The use of these formulas provides a fairly quick solution to various mathematical problems, and also helps to simplify expressions. The rules of algebraic transformations allow you to perform some manipulations with expressions, following which you can get the expression on the left side of the equality that is on the right side, or transform the right side of the equality (to get the expression on the left side after the equal sign).

It is convenient to know the formulas used for abbreviated multiplication by memory, as they are often used in solving problems and equations. The main formulas included in this list and their names are listed below.

sum square

To calculate the square of the sum, you need to find the sum consisting of the square of the first term, twice the product of the first term and the second, and the square of the second. In the form of an expression, this rule is written as follows: (a + c)² = a² + 2ac + c².

The square of the difference

To calculate the square of the difference, you need to calculate the sum consisting of the square of the first number, twice the product of the first number by the second (taken with the opposite sign), and the square of the second number. In the form of an expression, this rule looks like this: (a - c)² \u003d a² - 2ac + c².

Difference of squares

The formula for the difference of two numbers squared is equal to the product of the sum of these numbers and their difference. In the form of an expression, this rule looks like this: a² - c² \u003d (a + c) (a - c).

sum cube

To calculate the cube of the sum of two terms, it is necessary to calculate the sum consisting of the cube of the first term, triple the product of the square of the first term and the second, the triple product of the first term and the second squared, and the cube of the second term. In the form of an expression, this rule looks like this: (a + c)³ \u003d a³ + 3a²c + 3ac² + c³.

Sum of cubes

According to the formula, it is equal to the product of the sum of these terms and their incomplete square of the difference. In the form of an expression, this rule looks like this: a³ + c³ \u003d (a + c) (a² - ac + c²).

Example. It is necessary to calculate the volume of the figure, which is formed by adding two cubes. Only the magnitudes of their sides are known.

If the values of the sides are small, then it is easy to perform calculations.

If the lengths of the sides are expressed in cumbersome numbers, then in this case it is easier to apply the "Sum of Cubes" formula, which will greatly simplify the calculations.

difference cube

The expression for the cubic difference sounds like this: as the sum of the third power of the first term, triple the negative product of the square of the first term by the second, triple the product of the first term by the square of the second, and the negative cube of the second term. In the form of a mathematical expression, the difference cube looks like this: (a - c)³ \u003d a³ - 3a²c + 3ac² - c³.

Difference of cubes

The formula for the difference of cubes differs from the sum of cubes by only one sign. Thus, the difference of cubes is a formula equal to the product of the difference of these numbers by their incomplete square of the sum. In the form, the difference of cubes looks like this: a 3 - c 3 \u003d (a - c) (a 2 + ac + c 2).

Example. It is necessary to calculate the volume of the figure that will remain after subtracting the yellow volumetric figure, which is also a cube, from the volume of the blue cube. Only the size of the side of a small and large cube is known.

If the values of the sides are small, then the calculations are quite simple. And if the lengths of the sides are expressed in significant numbers, then it is worth using a formula entitled "Difference of Cubes" (or "Difference Cube"), which will greatly simplify the calculations.

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 represents identity. This explains why abbreviated multiplication formulas are also called abbreviated multiplication identities.

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

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 the implementation 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 .

Solution.

In this expression, squaring can be performed 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.

Abbreviated multiplication formulas. Training.

Try to calculate the following expressions in this way:

Answers:

Or, if you know the squares of basic two-digit numbers, remember how much will it be? Remembered? . Great! Since we are squaring, we must multiply by. It turns out that.

Remember that the formulas for the square of the sum and the square of the difference are valid not only for numerical expressions:

Calculate the following expressions for yourself:

Answers:

Abbreviated multiplication formulas. Outcome.

Let's sum up a little and write down the formulas for the square of the sum and difference in one line:

Now let's practice "assembling" the formula from the expanded view to the view. We will need this skill in the future when converting large expressions.

Let's say we have the following expression:

We know that the square of the sum (or difference) is square of one number square of another number And double the product of these numbers.

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

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

Where is the second number in our bracket.

The second number inside the parenthesis is equal to.

Let's check. should be equal. Indeed, it is, which means that we have found both numbers present in brackets: and. It remains to determine the sign that stands between them. What do you think the sign will be?

Right! Since we add double the product, then there will be an addition sign between the numbers. Now write down the transformed expression. Did you manage? You should get the following:

Note: changing the places of the terms does not affect the result (it does not matter if the addition or subtraction is between and).

It is not necessary that the terms in the expression to be converted stand as written in the formula. Look at this expression: . Try to convert it yourself. Happened?

Practice - transform the following expressions:

Answers: Did you manage? Let's fix the topic. Choose from the expressions below those that can be represented as a squared sum or difference.

- prove that it is equivalent.

- cannot be represented as a square; one could imagine if instead there was.

Difference of squares

Another formula for abbreviated multiplication is the difference of squares.

The difference of squares is not the square of the difference!

The difference of the squares of two numbers is equal to the product of the sum of these numbers and their difference:

Let's check if this formula is correct. To do this, we multiply, as we did when deriving the formulas for the square of the sum and difference:

Thus, we have just verified that the formula is indeed correct. This formula also simplifies complex calculations. Here's an example:

It is necessary to calculate: . Sure, we can square, then square, and subtract one from the other, but the formula makes it easy for us:

Happened? Let's check the results:

Just like the square of the sum (difference), the difference of squares formula can be applied not only to numbers:

The ability to decompose the difference of squares will help us transform complex mathematical expressions.

Pay attention:

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

Be careful and see which particular term is squared! To fix the topic, transform the following expressions:

Recorded? Let's compare the resulting expressions:

Now that you have mastered the square of the sum and the square of the difference, as well as the difference of squares, let's try to solve examples for a combination of these three formulas.

Transformation of elementary expressions (square of sum, square of difference, difference of squares)

Suppose we are given an example

This expression needs to be simplified. Look closely, what do you see in the numerator? That's right, the numerator is a perfect square:

When simplifying an expression, remember that the clue in which direction to move in the simplification is in the denominator (or numerator). In our case, when the denominator is decomposed, and nothing else can be done, we can understand that the numerator will be either the square of the sum or the square of the difference. Since we are adding, it becomes clear that the numerator is the square of the sum.

Try to convert the following expressions yourself:

Happened? Compare answers and move on!

Sum Cube and Difference Cube

The sum cube and difference cube formulas are derived in the same way as sum squared And difference squared: opening brackets when multiplying terms with each other.

If the square of the sum and the square of the difference are very easy to remember, then the question arises “how to remember the cubes?”

Look carefully at the two formulas described in comparison with the squaring of similar terms:

What pattern do you see?

1. When erecting in square we have square first number and square second; when cubed - yes cube one number and cube another number.

2. When erected in square, we have double product of numbers (numbers to the power of 1, which is one power less than the one to which we raise the expression); when erected in cube - tripled a product in which one of the numbers is squared (which is also 1 power less than the power to which we raise the expression).

3. When squaring, the sign in brackets in the expanded expression is reflected when adding (or subtracting) the double product - if addition is in brackets, then we add, if subtraction - subtract; when cubed, the rule is this: if we have a sum cube, then all the signs are “+”, and if we have a difference cube, then the signs alternate: “” - “” - “” - “”.

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

Shall we practice? Expand the brackets in the following expressions:

Compare the resulting expressions:

Difference and sum of cubes

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

As we remember, in the difference of squares, we are multiplying the difference and the sum of these numbers one by one. There are also two brackets in the difference of cubes and in the sum of cubes:

1 bracket - the difference (or sum) of numbers to the first degree (depending on whether we reveal the difference or sum of cubes);

2nd bracket - an incomplete square (look closely: if we subtracted (or added) the double product of numbers, there would be a square), the sign when multiplying numbers is opposite to the sign of the original expression.

To reinforce the topic, let's solve a few examples:

Compare the resulting expressions:

Training

Answers:

Let's summarize:

There are 7 abbreviated multiplication formulas:

ADVANCED LEVEL

Abbreviated multiplication formulas are formulas, knowing which you can avoid performing some standard operations when simplifying expressions or factoring polynomials. You need to know the formulas for abbreviated multiplication by heart!

sum square two expressions is equal to the square of the first expression plus twice the product of the first expression and the second plus the square of the second expression:
The square of the difference two expressions is equal to the square of the first expression minus twice the product of the first expression and the second plus the square of the second expression:
Difference of squares two expressions is equal to the product of the difference of these expressions and their sum:
sum cube two expressions is equal to the cube of the first expression plus three times the square of the first expression times the second plus three times the product of the first expression times the square of the second plus the cube of the second expression:
difference cube two expressions is equal to the cube of the first expression minus three times the product of the square of the first expression by the second plus three times the product of the first expression and the square of the second minus the cube of the second expression:
Sum of cubes two expressions is equal to the product of the sum of the first and second expressions by the incomplete square of the difference of these expressions:
Difference of cubes two expressions is equal to the product of the difference of the first and second expressions by the incomplete square of the sum of these expressions: