Formulas for simplified multiplication. Online calculator. Polynomial simplification. Polynomial multiplication

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”.

>>Math: Reduced multiplication formulas

Abbreviated multiplication formulas

There are several cases where the multiplication of one polynomial by another leads to a compact, easy-to-remember result. In these cases, it is preferable not to multiply one time each time polynomial on the other, and use the finished result. Let's consider these cases.

1. The square of the sum and the square of the difference:

Example 1 Open brackets in an expression:

a) (3x + 2) 2 ;

b) (5a 2 - 4b 3) 2

a) We use formula (1), taking into account that the role of a is played by 3x, and the role of b is the number 2.
We get:

(Zx + 2) 2 = (3x) 2 + 2 Zx 2 + 2 2 = 9x 2 + 12x + 4.

b) We use formula (2), considering that in the role a speaks 5a 2, and in the role b speaks 4b 3. We get:

(5a 2 -4b 3) 2 \u003d (5a 2) 2 - 2- 5a 2 4b 3 + (4b 3) 2 \u003d 25a 4 -40a 2 b 3 + 16b 6.

When using the formulas for the square of the sum or the square of the difference, keep in mind that
(- a - b) 2 \u003d (a + b) 2;
(b-a) 2 = (a-b) 2 .

This follows from the fact that (- a) 2 = a 2 .

Note that some mathematical tricks are based on formulas (1) and (2), allowing you to make calculations in your head.

For example, one can practically verbally square numbers ending in 1 and 9. Indeed

71 2 = (70 + 1) 2 = 70 2 + 2 70 1 + 1 2 = 4900 + 140 + 1 = 5041;
91 2 = (90 + I) 2 = 90 2 + 2 90 1 + 1 2 = 8100 + 180 + 1 = 8281;
69 2 \u003d (70 - I) 2 \u003d 70 2 - 2 70 1 + 1 2 \u003d 4900 - 140 + 1 \u003d 4761.

Sometimes you can also quickly square a number ending in 2 or 8. For example,

102 2 = (100 + 2) 2 = 100 2 + 2 100 2 + 2 2 = 10 000 + 400 + 4 = 10 404;

48 2 = (50 - 2) 2 = 50 2 - 2 50 2 + 2 2 = 2500 - 200 + 4 = 2304.

But the most elegant trick involves squaring numbers that end in 5.
Let us carry out the corresponding reasoning for 85 2 .

We have:

85 2 = (80 + 5) 2 = 80 2 + 2 80 5 + 5 2 =-80 (80+ 10)+ 25 = 80 90 + 25 = 7200 + 25 = 7225.

We note that to calculate 85 2 it was enough to multiply 8 by 9 and add 25 to the right to the result obtained. Similarly, you can do the same in other cases. For example, 35 2 \u003d 1225 (3 4 \u003d 12 and 25 was added to the resulting number on the right);

65 2 = 4225; 1252 \u003d 15625 (12 18 \u003d 156 and 25 was added to the resulting number on the right).

Since we are talking about various curious circumstances associated with boring (at first glance) formulas (1) and (2), we will supplement this conversation with the following geometric reasoning. Let a and b be positive numbers. Consider a square with side a + b and cut out squares with sides equal to a and b, respectively, in two of its corners (Fig. 4).

The area of a square with side a + b is (a + b) 2 . But we cut this square into four parts: a square with side a (its area is a 2), a square with side b (its area is b 2), two rectangles with sides a and b (the area of each such rectangle is ab). Hence, (a + b) 2 = a 2 + b 2 + 2ab, i.e., we have obtained formula (1).

Multiply the binomial a + b by the binomial a - b. We get:
(a + b) (a - b) \u003d a 2 - ab + ba - b 2 \u003d a 2 - b 2.
So

Any equality in mathematics is used both from left to right (i.e. the left side of the equality is replaced by its right side) and from right to left (i.e. the right side of the equality is replaced by its left side). If formula C) is used from left to right, then it allows you to replace the product (a + b) (a - b) with the finished result a 2 - b 2 . The same formula can be used from right to left, then it allows you to replace the difference of squares a 2 - b 2 with the product (a + b) (a - b). Formula (3) in mathematics is given a special name - the difference of squares.

Comment. Do not confuse the terms "difference of squares" to and "squared difference". The difference of squares is a 2 - b 2, which means that we are talking about formula (3); the square of the difference is (a-b) 2, so we are talking about formula (2). In ordinary language, formula (3) is read "from right to left" as follows:

the difference of the squares of two numbers (expressions) is equal to the product of the sum of these numbers (expressions) and their difference,

Example 2 Perform multiplication

(3x-2y)(3x+2y)
Solution. We have:
(3x - 2y) (3x + 2y) \u003d (3x) 2 - (2y) 2 \u003d 9x 2 - 4y 2.

Example 3 Express the binomial 16x 4 - 9 as a product of binomials.

Solution. We have: 16x 4 \u003d (4x 2) 2, 9 \u003d Z 2, which means that the given binomial is the difference of squares, i.e. formula (3), read from right to left, can be applied to it. Then we get:

16x 4 - 9 = (4x 2) 2 - W 2 = (4x 2 + 3)(4x 2 - 3)

Formula (3), like formulas (1) and (2), is used for mathematical tricks. See:

79 81 = (80 - 1) (80 + 1) - 802 - I2 = 6400 - 1 = 6399;
42 38 = D0 + 2) D0 - 2) = 402 - 22 = 1600 - 4 = 1596.

Let's finish the conversation about the formula for the difference of squares with a curious geometric reasoning. Let a and b be positive numbers, where a > b. Consider a rectangle with sides a + b and a - b (Fig. 5). Its area is (a + b) (a - b). Let's cut off a rectangle with sides b and a - b and glue it to the remaining part as shown in Figure 6. It is clear that the resulting figure has the same area, i.e. (a + b) (a - b). But this figure can
build like this: from a square with side a, cut out a square with side b (this is clearly seen in Fig. 6). So the area of the new figure is a 2 - b 2 . So, (a + b) (a - b) \u003d a 2 - b 2, i.e., we got the formula (3).

3. Difference of cubes and sum of cubes

Multiply the binomial a - b by the trinomial a 2 + ab + b 2.
We get:
(a - b) (a 2 + ab + b 2) \u003d a a 2 + a ab + a b 2 - b a 2 - b ab -b b 2 \u003d a 3 + a 2 b + ab 2 -a 2 b- ab 2 -b 3 \u003d a 3 -b 3.

Similarly

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

(check it yourself). So,

Formula (4) is usually called difference of cubes, formula (5) - the sum of cubes. Let's try to translate formulas (4) and (5) into ordinary language. Before doing this, note that the expression a 2 + ab + b 2 is similar to the expression a 2 + 2ab + b 2 that appeared in formula (1) and gave (a + b) 2 ; the expression a 2 - ab + b 2 is similar to the expression a 2 - 2ab + b 2 that appeared in formula (2) and gave (a - b) 2 .

To distinguish (in the language) these pairs of expressions from each other, each of the expressions a 2 + 2ab + b 2 and a 2 - 2ab + b 2 is called a perfect square (sum or difference), and each of the expressions a 2 + ab + b 2 and a 2 - ab + b 2 is called an incomplete square (sum or difference). Then we get the following translation of formulas (4) and (5) (read "from right to left") into ordinary language:

the difference of cubes of two numbers (expressions) is equal to the product of the difference of these numbers (expressions) by the incomplete square of their sum; the sum of cubes of two numbers (expressions) is equal to the product of the sum of these numbers (expressions) by the incomplete square of their difference.

Comment. All formulas (1)-(5) obtained in this section are used both from left to right and from right to left, only in the first case (from left to right) they say that (1)-(5) are reduced multiplication formulas, and in the second case (from right to left) they say that (1)-(5) are factorization formulas.

Example 4 Multiply (2x-1)(4x2 + 2x+1).

Solution. Since the first factor is the difference between the monomials 2x and 1, and the second factor is the incomplete square of their sum, then formula (4) can be used. We get:

(2x - 1) (4x 2 + 2x + 1) \u003d (2x) 3 - I 3 \u003d 8x 3 - 1.

Example 5 Express the binomial 27a 6 + 8b 3 as a product of polynomials.

Solution. We have: 27а 6 = (For 2) 3 , 8b 3 = (2b) 3 . This means that the given binomial is the sum of cubes, i.e., formula 95) can be applied to it, read from right to left. Then we get:

27a 6 + 8b 3 = (For 2) 3 + (2b) 3 = (For 2 + 2b) ((For 2) 2 - For 2 2b + (2b) 2) = (For 2 + 2b) (9a 4 - 6a 2 b + 4b 2).

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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 deciding practical examples often use abbreviated multiplication formulas 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 special case Newton's binomial formulas 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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