When calculating algebraic polynomials, to simplify calculations, we use abbreviated multiplication formulas . There are seven such formulas in total. They all need to be known by heart.

It should also be remembered that instead of a and b in formulas, there can be both numbers and any other algebraic polynomials.

Difference of squares

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

a 2 - b 2 = (a - b) (a + b)

sum square

The square of the sum of two numbers is equal to the square of the first number plus twice the product of the first number and the second plus the square of the second number.

(a + b) 2 = a 2 + 2ab + b 2

Note that with this reduced multiplication formula, it is easy to find the squares of large numbers without using a calculator or long multiplication. Let's explain with an example:

Find 112 2 .

Let us decompose 112 into the sum of numbers whose squares we remember well.2
112 = 100 + 1

We write the sum of numbers in brackets and put a square over the brackets.
112 2 = (100 + 12) 2

Let's use the sum square formula:
112 2 = (100 + 12) 2 = 100 2 + 2 x 100 x 12 + 12 2 = 10,000 + 2,400 + 144 = 12,544

Remember that the square sum formula is also valid for any algebraic polynomials.

(8a + c) 2 = 64a 2 + 16ac + c 2

Warning!!!

(a + b) 2 not equal to a 2 + b 2

The square of the difference

The square of the difference between two numbers is equal to the square of the first number minus twice the product of the first and the second plus the square of the second number.

(a - b) 2 = a 2 - 2ab + b 2

It is also worth remembering a very useful transformation:

(a - b) 2 = (b - a) 2
The formula above is proved by simply expanding the parentheses:

(a - b) 2 = a 2 - 2ab + b 2 = b 2 - 2ab + a 2 = (b - a) 2

sum cube

The cube of the sum of two numbers is equal to the cube of the first number plus three times the square of the first number times the second plus three times the product of the first times the square of the second plus the cube of the second.

(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3

Remembering this "terrible"-looking formula is quite simple.

Learn that a 3 comes first.

The two polynomials in the middle have coefficients of 3.

ATremember that any number to the zero power is 1. (a 0 = 1, b 0 = 1). It is easy to see that in the formula there is a decrease in the degree a and an increase in the degree b. You can verify this:
(a + b) 3 = a 3 b 0 + 3a 2 b 1 + 3a 1 b 2 + b 3 a 0 = a 3 + 3a 2 b + 3ab 2 + b 3

Warning!!!

(a + b) 3 not equal to a 3 + b 3

difference cube

The cube of the difference between two numbers is equal to the cube of the first number minus three times the square of the first number and the second plus three times the product of the first number and the square of the second minus the cube of the second.

(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3

This formula is remembered as the previous one, but only taking into account the alternation of the signs "+" and "-". The first member of a 3 is preceded by a "+" (according to the rules of mathematics, we do not write it). This means that the next member will be preceded by "-", then again "+", etc.

(a - b) 3 = + a 3 - 3a 2b + 3ab 2 - b 3 \u003d a 3 - 3a 2 b + 3ab 2 - b 3

The sum of cubes ( Not to be confused with the sum cube!)

The sum of cubes is equal to the product of the sum of two numbers and the incomplete square of the difference.

a 3 + b 3 = (a + b)(a 2 - ab + b 2)

The sum of cubes is the product of two brackets.

The first parenthesis is the sum of two numbers.

The second bracket is the incomplete square of the difference of numbers. The incomplete square of the difference is called the expression:

A 2 - ab + b 2
This square is incomplete, since in the middle, instead of a double product, there is an ordinary product of numbers.

Cube Difference (Not to be confused with Difference Cube!!!)

The difference of cubes is equal to the product of the difference of two numbers by the incomplete square of the sum.

a 3 - b 3 \u003d (a - b) (a 2 + ab + b 2)

Be careful when writing characters.It should be remembered that all the formulas above are also used from right to left.

An easy way to remember abbreviated multiplication formulas, or... Pascal's Triangle.

Is it difficult to remember the formulas of abbreviated multiplication? The case is easy to help. You just need to remember how such a simple thing as Pascal's triangle is depicted. Then you will remember these formulas always and everywhere, or rather, do not remember, but restore.

What is Pascal's Triangle? This triangle consists of the coefficients that enter into the expansion of any power of a binomial of the form into a polynomial.

Let's break it down, for example:

In this record, it is easy to remember that at the beginning there is a cube of the first, and at the end - the cube of the second number. But what's in the middle is hard to remember. And even the fact that in each next term the degree of one factor decreases all the time, and the second increases - it is easy to notice and remember, it is more difficult to remember the coefficients and signs (plus or minus?).

So, first the odds. You don't have to memorize them! On the margins of the notebook, we quickly draw Pascal's triangle, and here they are - the coefficients, already in front of us. We start drawing with three ones, one on top, two below, to the right and to the left - yeah, already a triangle is obtained:

The first line, with one one, is zero. Then comes the first, second, third and so on. To get the second line, you need to add ones again along the edges, and in the center write down the number obtained by adding the two numbers above it:

We write the third line: again along the edges of the unit, and again, to get the next number in a new line, add the numbers above it in the previous one:

As you may have guessed, we get in each line the coefficients from the decomposition of a binomial into a polynomial:

Well, it’s even easier to remember the signs: the first one is the same as in the expanded binomial (we lay out the sum - that means plus, the difference - that means minus), and then the signs alternate!

This is such a useful thing - Pascal's triangle. Enjoy!

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, which 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, you need 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 of a mathematical expression, 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 is an identity. This explains why abbreviated multiplication formulas are also called abbreviated multiplication identities.

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

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 performing 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 .

Decision.

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.

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In the previous lesson, we dealt with factorization. We mastered two methods: taking the common factor out of brackets and grouping. In this tutorial, the following powerful method: abbreviated multiplication formulas. In a short note - FSU.

Abbreviated multiplication formulas (square of sum and difference, cube of sum and difference, difference of squares, sum and difference of cubes) are essential in all branches of mathematics. They are used in simplifying expressions, solving equations, multiplying polynomials, reducing fractions, solving integrals, etc. etc. In short, there is every reason to deal with them. Understand where they come from, why they are needed, how to remember them and how to apply them.

Do we understand?)

Where do abbreviated multiplication formulas come from?

Equalities 6 and 7 are not written in a very usual way. Like the opposite. This is on purpose.) Any equality works both from left to right and from right to left. In such a record, it is clearer where the FSO comes from.

They are taken from multiplication.) For example:

(a+b) 2 =(a+b)(a+b)=a 2 +ab+ba+b 2 =a 2 +2ab+b 2

That's it, no scientific tricks. We just multiply the brackets and give similar ones. This is how it turns out all abbreviated multiplication formulas. abbreviated multiplication is because in the formulas themselves there is no multiplication of brackets and reduction of similar ones. Reduced.) The result is immediately given.

FSU needs to know by heart. Without the first three, you can not dream of a triple, without the rest - about a four with a five.)

Why do we need abbreviated multiplication formulas?

There are two reasons to learn, even memorize, these formulas. The first - a ready-made answer on the machine dramatically reduces the number of errors. But this is not the main reason. And here's the second one...

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.