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 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.
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”.
Abbreviated multiplication formulas (FSU) are used to exponentiate and multiply numbers and expressions. Often these formulas allow you to make calculations more compactly and quickly.
In this article, we will list the main formulas for abbreviated multiplication, group them into a table, consider examples of using these formulas, and also dwell on the principles for proving abbreviated multiplication formulas.
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For the first time, the topic of FSU is considered within the course "Algebra" for the 7th grade. Below are 7 basic formulas.
Abbreviated multiplication formulas
- sum square formula: a + b 2 = a 2 + 2 a b + b 2
- difference square formula: a - b 2 \u003d a 2 - 2 a b + b 2
- sum cube formula: a + b 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3
- difference cube formula: a - b 3 \u003d a 3 - 3 a 2 b + 3 a b 2 - b 3
- difference of squares formula: a 2 - b 2 \u003d a - b a + b
- formula for the sum of cubes: a 3 + b 3 \u003d a + b a 2 - a b + b 2
- cube difference formula: a 3 - b 3 \u003d a - b a 2 + a b + b 2
The letters a, b, c in these expressions can be any numbers, variables or expressions. For ease of use, it is better to learn the seven basic formulas by heart. We summarize them in a table and give them below, circling them with a box.
The first four formulas allow you to calculate, respectively, the square or cube of the sum or difference of two expressions.
The fifth formula calculates the difference of squares of expressions by multiplying their sum and difference.
The sixth and seventh formulas are, respectively, the multiplication of the sum and difference of expressions by the incomplete square of the difference and the incomplete square of the sum.
The abbreviated multiplication formula is sometimes also called the abbreviated multiplication identities. This is not surprising, since every equality is an identity.
When solving practical examples, abbreviated multiplication formulas are often used with rearranged left and right parts. This is especially convenient when factoring a polynomial.
Additional abbreviated multiplication formulas
We will not limit ourselves to the 7th grade course in algebra and add a few more formulas to our FSU table.
First, consider Newton's binomial formula.
a + b n = C n 0 a n + C n 1 a n - 1 b + C n 2 a n - 2 b 2 + . . + C n n - 1 a b n - 1 + C n n b n
Here C n k are the binomial coefficients that are in line number n in pascal's triangle. Binomial coefficients are calculated by the formula:
C nk = n ! k! · (n - k) ! = n (n - 1) (n - 2) . . (n - (k - 1)) k !
As you can see, the FSU for the square and cube of the difference and the sum is a special case of Newton's binomial formula for n=2 and n=3, respectively.
But what if there are more than two terms in the sum to be raised to a power? The formula for the square of the sum of three, four or more terms will be useful.
a 1 + a 2 + . . + a n 2 = a 1 2 + a 2 2 + . . + a n 2 + 2 a 1 a 2 + 2 a 1 a 3 + . . + 2 a 1 a n + 2 a 2 a 3 + 2 a 2 a 4 + . . + 2 a 2 a n + 2 a n - 1 a n
Another formula that may come in handy is the formula for the difference of the nth powers of two terms.
a n - b n = a - b a n - 1 + a n - 2 b + a n - 3 b 2 + . . + a 2 b n - 2 + b n - 1
This formula is usually divided into two formulas - respectively for even and odd degrees.
For even exponents 2m:
a 2 m - b 2 m = a 2 - b 2 a 2 m - 2 + a 2 m - 4 b 2 + a 2 m - 6 b 4 + . . + b 2 m - 2
For odd exponents 2m+1:
a 2 m + 1 - b 2 m + 1 = a 2 - b 2 a 2 m + a 2 m - 1 b + a 2 m - 2 b 2 + . . + b 2 m
The formulas for the difference of squares and the difference of cubes, you guessed it, are special cases of this formula for n = 2 and n = 3, respectively. For the difference of cubes, b is also replaced by - b .
How to read abbreviated multiplication formulas?
We will give the corresponding formulations for each formula, but first we will deal with the principle of reading formulas. The easiest way to do this is with an example. Let's take the very first formula for the square of the sum of two numbers.
a + b 2 = a 2 + 2 a b + b 2 .
They say: the square of the sum of two expressions a and b is equal to the sum of the square of the first expression, twice the product of the expressions and the square of the second expression.
All other formulas are read similarly. For the squared difference a - b 2 \u003d a 2 - 2 a b + b 2 we write:
the square of the difference of two expressions a and b is equal to the sum of the squares of these expressions minus twice the product of the first and second expressions.
Let's read the formula a + b 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3. The cube of the sum of two expressions a and b is equal to the sum of the cubes of these expressions, three times the product of the square of the first expression and the second, and three times the product of the square of the second expression and the first expression.
We proceed to reading the formula for the difference of cubes a - b 3 \u003d a 3 - 3 a 2 b + 3 a b 2 - b 3. The cube of the difference of two expressions a and b is equal to the cube of the first expression minus three times the square of the first expression and the second, plus three times the square of the second expression and the first expression, minus the cube of the second expression.
The fifth formula a 2 - b 2 \u003d a - b a + b (difference of squares) reads like this: the difference of the squares of two expressions is equal to the product of the difference and the sum of the two expressions.
Expressions like a 2 + a b + b 2 and a 2 - a b + b 2 for convenience are called, respectively, the incomplete square of the sum and the incomplete square of the difference.
With this in mind, the formulas for the sum and difference of cubes are read as follows:
The sum of the cubes of two expressions is equal to the product of the sum of these expressions and the incomplete square of their difference.
The difference of the cubes of two expressions is equal to the product of the difference of these expressions by the incomplete square of their sum.
FSU Proof
Proving FSU is quite simple. Based on the properties of multiplication, we will carry out the multiplication of the parts of the formulas in brackets.
For example, consider the formula for the square of the difference.
a - b 2 \u003d a 2 - 2 a b + b 2.
To raise an expression to the second power, the expression must be multiplied by itself.
a - b 2 \u003d a - b a - b.
Let's expand the brackets:
a - b a - b \u003d a 2 - a b - b a + b 2 \u003d a 2 - 2 a b + b 2.
The formula has been proven. The other FSOs are proved similarly.
Examples of application of FSO
The purpose of using reduced multiplication formulas is to quickly and concisely multiply and exponentiate expressions. However, this is not the entire scope of the FSO. They are widely used in reducing expressions, reducing fractions, factoring polynomials. Let's give examples.
Example 1. FSO
Let's simplify the expression 9 y - (1 + 3 y) 2 .
Apply the sum of squares formula and get:
9 y - (1 + 3 y) 2 = 9 y - (1 + 6 y + 9 y 2) = 9 y - 1 - 6 y - 9 y 2 = 3 y - 1 - 9 y 2
Example 2. FSO
Reduce the fraction 8 x 3 - z 6 4 x 2 - z 4 .
We notice that the expression in the numerator is the difference of cubes, and in the denominator - the difference of squares.
8 x 3 - z 6 4 x 2 - z 4 \u003d 2 x - z (4 x 2 + 2 x z + z 4) 2 x - z 2 x + z.
We reduce and get:
8 x 3 - z 6 4 x 2 - z 4 = (4 x 2 + 2 x z + z 4) 2 x + z
FSUs also help to calculate the values of expressions. The main thing is to be able to notice where to apply the formula. Let's show this with an example.
Let's square the number 79. Instead of cumbersome calculations, we write:
79 = 80 - 1 ; 79 2 = 80 - 1 2 = 6400 - 160 + 1 = 6241 .
It would seem that a complex calculation was carried out quickly with just the use of abbreviated multiplication formulas and a multiplication table.
Another important point- selection of the square of the binomial. The expression 4 x 2 + 4 x - 3 can be converted to 2 x 2 + 2 2 x 1 + 1 2 - 4 = 2 x + 1 2 - 4 . Such transformations are widely used in integration.
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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, the situation is more difficult with remembering 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!
They are used to simplify calculations, as well as the decomposition of polynomials into factors, the rapid multiplication of polynomials. Most of the abbreviated multiplication formulas can be obtained from Newton's binomial - you will soon see this.
Formulas for squares often used in calculations. They begin to be studied in the school curriculum from the 7th grade until the end of the training, the formulas for squares and cubes, students should know by heart.
Cube formulas not very complex and they need to be known when reducing polynomials to a standard form, to simplify the rise of the sum or difference of a variable and a number to a cube.
Formulas marked in red are obtained from the previous grouping of similar terms.
Formulas for the fourth and fifth powers in the school course, few will be useful, however, there are tasks in the study of higher mathematics where you need to calculate the coefficients at the degrees.
Degree formulas n are painted in terms of binomial coefficients using factorials as follows 
Examples of application of abbreviated multiplication formulas
Example 1. Calculate 51^2.
Solution. If you have a calculator, you can easily find it
I was joking - everyone is wise with a calculator, without it ... (let's not talk about sad things).
Without a calculator and knowing the above rules, we find the square of the number by the rule
Example 2 Find 99^2.
Solution. Apply the second formula
Example 3: Squaring an expression
(x+y-3).
Solution. We mentally consider the sum of the first two terms as one term and, according to the second formula for abbreviated multiplication, we have
Example 4. Find the difference of squares
11^2-9^2.
Solution. Since the numbers are small, you can simply substitute the values of the squares
But our goal is completely different - to learn how to use abbreviated multiplication formulas to simplify calculations. For this example, apply the third formula
Example 5. Find the difference of squares
17^2-3^2
.
Solution. In this example, you will already want to learn the rules to reduce the calculations to one line
As you can see, we didn't do anything amazing.
Example 6: Simplify an expression
(x-y)^2-(x+y)^2.
Solution. You can lay out the squares, and later group like terms. However, one can directly apply the difference of squares 
Simple and without long solutions.
Example 7. Cube a polynomial
x^3-4.
Solution . Let's apply the 5 abbreviated multiplication formula 
Example 8. Write as a difference of squares or their sum
a) x^2-8x+7
b) x^2+4x+29
Solution. a) Rearrange the terms
b) Simplify based on the previous reasoning 
Example 9. Expand a rational fraction
Solution. Apply the difference of squares formula 
We compose a system of equations for determining the constants
We add the second equation to the tripled first equation. We substitute the found value into the first equation 
Finally, the expansion takes the form 
It is often necessary to expand a rational fraction before integrating in order to reduce the power of the denominator.
Example 10. Using Newton's binomial, paint
expression (x-a)^7.
Solution. You probably already know what Newton's binomial is. If not, then below are the binomial coefficients
They are formed as follows: there are units along the edge, the coefficients between them in the bottom line are formed by summing the neighboring upper ones. If we are looking for a difference to some extent, then the signs in the schedule alternate from plus to minus. Thus, for the seventh order, we get the following alignment
Carefully also look at how the indicators change - for the first variable they decrease by one in each next term, respectively, for the second - they increase by one. In sum, the indicators should always be equal to the degree of decomposition (= 7).
I think on the basis of the above material you will be able to solve problems on Newton's binomial. Learn abbreviated multiplication formulas and apply wherever it can simplify calculations and save time on the task.
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...
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