Difference of squares
Let us derive the formula for the difference of the squares $ a ^ 2-b ^ 2 $.
To do this, remember the following rule:
If we add any monomial to the expression and subtract the same monomial, then we get the correct identity.
Add to our expression and subtract from it the monomial $ ab $:
Total, we get:
That is, the difference between the squares of two monomials is equal to the product of their difference by their sum.
Example 1
Represent as product $ (4x) ^ 2-y ^ 2 $
\ [(4x) ^ 2-y ^ 2 = ((2x)) ^ 2-y ^ 2 \]
\ [((2x)) ^ 2-y ^ 2 = \ left (2x-y \ right) (2x + y) \]
Sum of cubes
We derive the formula for the sum of cubes $ a ^ 3 + b ^ 3 $.
Factor out the common factors:
Let's take out $ \ left (a + b \ right) $ outside the brackets:
Total, we get:
That is, the sum of the cubes of two monomials is equal to the product of their sum by the incomplete square of their difference.
Example 2
Represent as a product $ (8x) ^ 3 + y ^ 3 $
This expression can be rewritten as follows:
\ [(8x) ^ 3 + y ^ 3 = ((2x)) ^ 3 + y ^ 3 \]
Using the formula for the difference of squares, we get:
\ [((2x)) ^ 3 + y ^ 3 = \ left (2x + y \ right) (4x ^ 2-2xy + y ^ 2) \]
Difference of cubes
Let us derive the formula for the difference of cubes $ a ^ 3-b ^ 3 $.
For this, we will use the same rule as above.
Add to our expression and subtract from it the monomials $ a ^ 2b \ and \ (ab) ^ 2 $:
Factor out the common factors:
Let's take out $ \ left (a-b \ right) $ outside the brackets:
Total, we get:
That is, the difference between the cubes of two monomials is equal to the product of their difference by the incomplete square of their sum.
Example 3
Represent as product $ (8x) ^ 3-y ^ 3 $
This expression can be rewritten as follows:
\ [(8x) ^ 3-y ^ 3 = ((2x)) ^ 3-y ^ 3 \]
Using the formula for the difference of squares, we get:
\ [((2x)) ^ 3-y ^ 3 = \ left (2x-y \ right) (4x ^ 2 + 2xy + y ^ 2) \]
An example of problems using the formulas for the difference of squares and the sum and difference of cubes
Example 4
Factor.
a) $ ((a + 5)) ^ 2-9 $
c) $ -x ^ 3 + \ frac (1) (27) $
Solution:
a) $ ((a + 5)) ^ 2-9 $
\ [(((a + 5)) ^ 2-9 = (a + 5)) ^ 2-3 ^ 2 \]
Applying the formula for the difference of squares, we get:
\ [((a + 5)) ^ 2-3 ^ 2 = \ left (a + 5-3 \ right) \ left (a + 5 + 3 \ right) = \ left (a + 2 \ right) (a +8) \]
Let's write this expression in the form:
Let's apply the formula of kuma cubes:
c) $ -x ^ 3 + \ frac (1) (27) $
Let's write this expression in the form:
\ [- x ^ 3 + \ frac (1) (27) = (\ left (\ frac (1) (3) \ right)) ^ 3-x ^ 3 \]
Let's apply the formula of kuma cubes:
\ [(\ left (\ frac (1) (3) \ right)) ^ 3-x ^ 3 = \ left (\ frac (1) (3) -x \ right) \ left (\ frac (1) ( 9) + \ frac (x) (3) + x ^ 2 \ right) \]
In the previous lessons, we looked at two ways to factor a polynomial into factors: parentheses and grouping.
In this lesson, we will look at another way to factorize a polynomial using abbreviated multiplication formulas.
We recommend prescribing each formula at least 12 times. For better memorization, write out all the formulas for abbreviated multiplication for yourself on a small cheat sheet.
Let's remember what the formula for the difference of cubes looks like.
a 3 - b 3 = (a - b) (a 2 + ab + b 2)The formula for the difference between cubes is not very easy to memorize, so we recommend using a special way to memorize it.
It is important to understand that any formula for abbreviated multiplication also works in reverse side.
(a - b) (a 2 + ab + b 2) = a 3 - b 3Let's look at an example. It is necessary to factor the difference between the cubes.
Note that "27a 3" is "(3a) 3", which means that for the formula for the difference between cubes, instead of "a" we use "3a".
We use the formula for the difference of cubes. In place "a 3" we have "27a 3", and in place "b 3", as in the formula, there is "b 3".
Applying the difference of cubes in the opposite direction
Let's look at another example. You want to convert the product of polynomials to the difference of cubes using the abbreviated multiplication formula.
Note that the product of polynomials "(x - 1) (x 2 + x + 1)" resembles the right side of the formula for the difference between cubes "", only instead of "a" there is "x", and instead of "b" there is "1" ...
We use for "(x - 1) (x 2 + x + 1)" the formula for the difference of cubes in the opposite direction.
Let's look at a more complicated example. It is required to simplify the product of polynomials.
If we compare "(y 2 - 1) (y 4 + y 2 + 1)" with the right side of the cubes difference formula
« a 3 - b 3 = (a - b) (a 2 + ab + b 2)", Then you can understand that in the place" a "from the first bracket is" y 2, and in place "b" is "1".
Formulas or abbreviated multiplication rules are used in arithmetic, or rather in algebra, for a faster process of calculating large algebraic expressions. The formulas themselves are derived from the rules existing in algebra for multiplying several polynomials.
The use of these formulas provides a fairly prompt solution to various mathematical problems, and also helps to simplify expressions. Rules algebraic transformations allow you to perform some manipulations with expressions, following which you can get the expression on the left side of the equality 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 reduced multiplication by memory, since they are often used in solving problems and equations. Below are the main formulas included in this list and their name.
Sum squared
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 by the second, and the square of the second. As an expression, this rule is written as follows: (a + c) ² = a² + 2ac + c².
Difference squared
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. As an expression, this rule looks like this: (a - c) ² = a² - 2ac + c².
Difference of squares
The formula for the difference between two numbers squared is equal to the product of the sum of these numbers by their difference. In the form of an expression, this rule looks as follows: a² - c² = (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, the triple product of the square of the first term and the second, triple product of the first term and the second squared, as well as the cube of the second term. In the form of an expression, this rule looks as follows: (a + c) ³ = a³ + 3a²c + 3ac² + c³.
Sum of cubes
According to the formula, it is equated to the product of the sum of these terms by their incomplete square of the difference. In the form of an expression, this rule looks as follows: a³ + c³ = (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 sizes of their sides are known.
If the side values are small, then the calculations are easy.
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 is as follows: 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 cube of the difference looks like this: (a - c) ³ = a³ - 3a²c + 3ac² - c³.
Difference of cubes
The difference between cubes formula differs from the sum of cubes in only one sign. Thus, the difference between the 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 as follows: a 3 - c 3 = (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 from the volume of the blue cube, which is also a cube. Only the size of the side of the small and large cube is known.
If the side values are small, then the calculations are fairly straightforward. And if the lengths of the sides are expressed in significant numbers, then it is worth using a formula entitled "Difference Cubes" (or "Difference Cube"), which will greatly simplify the calculations.
Abbreviated multiplication formulas.
Study of abbreviated multiplication formulas: the square of the sum and the square of the difference of two expressions; difference of squares of two expressions; the cube of the sum and the cube of the difference of two expressions; sum and difference of cubes of two expressions.
Application of abbreviated multiplication formulas when solving examples.
To simplify expressions, factorize polynomials, and bring polynomials to a standard form, abbreviated multiplication formulas are used. Abbreviated multiplication formulas need to be known by heart.
Let a, b R. Then:
1. The square of the sum of the two expressions is the square of the first expression plus twice the product of the first expression by the second plus the square of the second expression.
(a + b) 2 = a 2 + 2ab + b 2
2. The squared difference of the two expressions is the square of the first expression minus twice the product of the first expression by the second plus the square of the second expression.
(a - b) 2 = a 2 - 2ab + b 2
3. Difference of squares two expressions is equal to the product of the difference between these expressions and their sum.
a 2 - b 2 = (a -b) (a + b)
4. Sum cube of two expressions is equal to the cube of the first expression plus three times 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.
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
5. Difference cube two expressions 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 product of the first expression and the square of the second minus the cube of the second expression.
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
6. 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.
a 3 + b 3 = (a + b) (a 2 - ab + b 2)
7. 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.
a 3 - b 3 = (a - b) (a 2 + ab + b 2)
Application of abbreviated multiplication formulas when solving examples.
Example 1.
Calculate
a) Using the formula for the square of the sum of two expressions, we have
(40 + 1) 2 = 40 2 + 2 40 1 + 1 2 = 1600 + 80 + 1 = 1681
b) Using the formula for the square of the difference of two expressions, we get
98 2 = (100 - 2) 2 = 100 2 - 2 100 2 + 2 2 = 10000 - 400 + 4 = 9604
Example 2.
Calculate
Using the formula for the difference between the squares of the two expressions, we get
Example 3.
Simplify expression
(x - y) 2 + (x + y) 2
We use the formulas for the square of the sum and the square of the difference of two expressions
(x - y) 2 + (x + y) 2 = x 2 - 2xy + y 2 + x 2 + 2xy + y 2 = 2x 2 + 2y 2
Abbreviated multiplication formulas in one table:
(a + b) 2 = a 2 + 2ab + b 2
(a - b) 2 = a 2 - 2ab + b 2
a 2 - b 2 = (a - b) (a + b)
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
a 3 + b 3 = (a + b) (a 2 - ab + b 2)
a 3 - b 3 = (a - b) (a 2 + ab + b 2)
Abbreviated Multiplication Formulas (ACF) are used for exponentiation and multiplication of numbers and expressions. Often these formulas allow you to make calculations more compact and faster.
In this article, we will list the basic formulas for abbreviated multiplication, group them in a table, consider examples of using these formulas, and also dwell on the principles of proof of abbreviated multiplication formulas.
For the first time, the topic of FSU is considered within the framework of the "Algebra" course for the 7th grade. Below are 7 basic formulas.
Abbreviated multiplication formulas
- the formula for the square of the sum: a + b 2 = a 2 + 2 a b + b 2
- the formula for the square of the difference: a - b 2 = 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 = a 3 - 3 a 2 b + 3 a b 2 - b 3
- difference of squares formula: a 2 - b 2 = a - b a + b
- the formula for the sum of cubes: a 3 + b 3 = a + b a 2 - a b + b 2
- the formula for the difference of cubes: a 3 - b 3 = 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 best to learn the seven basic formulas by heart. Let's summarize them in a table and present them below, encircling them with a frame.
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 the squares of the expressions by the product of their sum and the difference.
The sixth and seventh formulas are, respectively, the multiplication of the sum and difference of expressions by an incomplete square of the difference and an 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 sides. This is especially useful when a factorization of a polynomial takes place.
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 the Newton 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 binomial coefficients that are in row n in the pascal triangle. Binomial coefficients are calculated by the formula:
C n k = n! k! (N - k)! = n (n - 1) (n - 2). ... (n - (k - 1)) k!
As you can see, the FSE for the square and the cube of the difference and the sum is a special case of the Newton 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 the 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 between the n-th 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 - for even and odd degrees, respectively.
For even indicators 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 with - b.
How to read abbreviated multiplication formulas?
We will give the appropriate formulations for each formula, but first we will understand the principle of reading formulas. The most convenient way to do this is by 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, the doubled product of the expressions and the square of the second expression.
All other formulas are read in the same way. For the square of the difference a - b 2 = a 2 - 2 a b + b 2 we write:
the square of the difference between the 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.
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 square of the first expression by the second, and three times the square of the second expression by the first expression.
We proceed to reading the formula for the difference between cubes a - b 3 = 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 = a - b a + b (difference of squares) reads as follows: 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 the cubes will read as follows:
The sum of the cubes of two expressions is equal to the product of the sum of these expressions by the incomplete square of their difference.
The difference between the cubes of two expressions is equal to the product of the difference between these expressions and the incomplete square of their sum.
Proof of FSO
It is quite easy to prove the FSO. Based on the properties of multiplication, we multiply the parts of the formulas in parentheses.
For example, consider the formula for the square of the difference.
a - b 2 = a 2 - 2 a b + b 2.
To raise an expression to the second power, you need to multiply this expression by itself.
a - b 2 = a - b a - b.
Let's expand the brackets:
a - b a - b = a 2 - a b - b a + b 2 = a 2 - 2 a b + b 2.
The formula is proven. The rest of the FSOs are proved in a similar way.
Examples of FSU application
The purpose of using abbreviated multiplication formulas is to multiply and exponentiate expressions quickly and concisely. However, this is not the entire scope of the FSO. They are widely used in abbreviating expressions, reducing fractions, factoring polynomials. Here are some examples.
Example 1. FSO
Simplify the expression 9 y - (1 + 3 y) 2.
We apply the formula for the sum of squares 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.
Note that the expression in the numerator is the difference between the cubes, and the denominator is the difference in the squares.
8 x 3 - z 6 4 x 2 - z 4 = 2 x - z (4 x 2 + 2 x z + z 4) 2 x - z 2 x + z.
We shorten and get:
8 x 3 - z 6 4 x 2 - z 4 = (4 x 2 + 2 x z + z 4) 2 x + z
FSOs 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 using the abbreviated multiplication formulas and the multiplication table.
Another important point is the 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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