In previous lessons, we looked at two ways to factor a polynomial: putting the common factor out of brackets and the grouping method.
In this lesson we will look at another way to factor a polynomial using abbreviated multiplication formulas.
We recommend that you write each formula at least 12 times. For better memorization write down all the formulas for abbreviated multiplication on a small cheat sheet.
Let's remember what the difference of cubes formula looks like.
a 3 − b 3 = (a − b)(a 2 + ab + b 2)The difference of cubes formula is not very easy to remember, so we recommend using a special method to remember it.
It is important to understand that any abbreviated multiplication formula 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 of cubes.
Please note that “27a 3” is “(3a) 3”, which means that for the difference of cubes formula, instead of “a” we use “3a”.
We use the difference of cubes formula. In place of “a 3” we have “27a 3”, and in place of “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 need to convert the product of polynomials into the difference of cubes using the abbreviated multiplication formula.
Please note that the product of polynomials “(x − 1)(x 2 + x + 1)” resembles the right side of the difference of cubes formula “”, only instead of “a” there is “x”, and in place of “b” there is “1” .
For “(x − 1)(x 2 + x + 1)” we use the difference of cubes formula 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 difference of cubes formula
« a 3 − b 3 = (a − b)(a 2 + ab + b 2)”, then you can understand that in place of “a” from the first bracket there is “y 2”, and in place of “b” there is “1”.
Abbreviated multiplication formulas or rules are used in arithmetic, more specifically algebra, to speed up the process of evaluating large algebraic expressions. The formulas themselves are derived from rules existing in algebra for multiplying 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 obtain on the left side of the equality the expression on the right side, or transform the right side of the equality (to obtain the expression on the left side after the equal sign).
It is convenient to know the formulas used for abbreviated multiplication from memory, since they are often used in solving problems and equations. Listed below are the main formulas included in this list, and their name.
Square of the sum
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.
Squared 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 and 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 = 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 - с² = (a + с)·(a - с).
Cube of sum
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, triple the 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 = 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 partial square differences. In the form of an expression, this rule looks like this: a³ + c³ = (a + c)·(a² - ac + c²).
Example. It is necessary to calculate the volume of a figure formed by adding two cubes. Only the sizes of their sides are known.
If the side values are small, then the calculations are simple.
If the lengths of the sides are expressed in cumbersome numbers, then in this case it is easier to use 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 cube of the difference looks like this: (a - c)³ = a³ - 3a²c + 3ac² - c³.
Difference of cubes
The difference of cubes formula 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 and their incomplete square of the sum. In the form, the difference of cubes looks like this: 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 volumetric figure from the volume of the blue cube yellow, which is also a cube. Only the side size of the small and large cube is known.
If the side values are small, then the calculations are quite simple. And if the lengths of the sides are expressed in significant numbers, then it is worth applying the formula entitled “Difference of cubes” (or “Cube of difference”), which will greatly simplify the calculations.
Abbreviated multiplication formulas (MMF) 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 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 formulas for abbreviated multiplication.
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
- formula for the square of the sum: a + b 2 = a 2 + 2 a b + b 2
- square difference formula: 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
- square difference formula: a 2 - b 2 = a - b a + b
- formula for sum of cubes: a 3 + b 3 = a + b a 2 - a b + b 2
- formula for 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 better to learn the seven basic formulas by heart. Let's put 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 between the squares of expressions by multiplying their sum and difference.
The sixth and seventh formulas are, respectively, multiplying 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 the left and right sides swapped. This is especially convenient when factoring a polynomial.
Additional abbreviated multiplication formulas
Let's not limit ourselves to the 7th grade algebra course and add a few more formulas to our FSU table.
First, let's look at 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 appear in line number n in Pascal’s triangle. Binomial coefficients are calculated using the formula:
C n k = 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 that needs 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 be useful is the formula for the difference between 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 - for even and odd powers, respectively.
For even 2m indicators:
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 difference of squares and difference of cubes formulas, as you guessed, are special cases of this formula for n = 2 and n = 3, respectively. For difference of cubes, b is also replaced by - 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 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 square of the difference a - b 2 = a 2 - 2 a b + b 2 we write:
the square of the difference between 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, triple the product of the square of the first expression by the second, and triple the product of the square of the second expression by the first expression.
Let's move on to reading the formula for the difference of cubes a - b 3 = a 3 - 3 a 2 b + 3 a b 2 - b 3. The cube of the difference between two expressions a and b is equal to the cube of the first expression minus the triple product of the square of the first expression and the second, plus the triple product of 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 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.
For convenience, expressions like a 2 + a b + b 2 and a 2 - a b + b 2 are called, respectively, the incomplete square of the sum and the incomplete square of the difference.
Taking this into account, the formulas for the sum and difference of cubes can be read as follows:
The sum of the cubes of two expressions is equal to the product of the sum of these expressions and the partial 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 partial square of their sum.
Proof of the FSU
Proving FSU is quite simple. Based on the properties of multiplication, we will multiply the parts of the formulas in brackets.
For example, consider the formula for the squared 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 remaining FSUs are proven similarly.
Examples of FSU application
The purpose of using abbreviated multiplication formulas is to quickly and concisely multiply and raise expressions to powers. However, this is not the entire scope of application of the FSU. They are widely used in reducing expressions, reducing fractions, and factoring polynomials. Let's give examples.
Example 1. FSU
Let's simplify the expression 9 y - (1 + 3 y) 2.
Let's 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. FSU
Let's reduce the fraction 8 x 3 - z 6 4 x 2 - z 4.
We note that the expression in the numerator is the difference of cubes, and in the denominator is the difference of 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 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 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, let's write:
79 = 80 - 1 ; 79 2 = 80 - 1 2 = 6400 - 160 + 1 = 6241 .
It would seem complex calculation carried out quickly just using abbreviated multiplication formulas and multiplication tables.
Another important point is the selection of the square of the binomial. The expression 4 x 2 + 4 x - 3 can be converted into 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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Abbreviated multiplication formulas.
Studying abbreviated multiplication formulas: the square of the sum and the square of the difference of two expressions; difference of squares of two expressions; cube of the sum and cube of the difference of two expressions; sums and differences of cubes of two expressions.
Application of abbreviated multiplication formulas when solving examples.
To simplify expressions, factor polynomials, and reduce polynomials to standard form, abbreviated multiplication formulas are used. Abbreviated multiplication formulas you need to know by heart.
Let a, b R. Then:
1. The square of the sum of 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.
(a + b) 2 = a 2 + 2ab + b 2
2. The square of the difference of 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.
(a - b) 2 = a 2 - 2ab + b 2
3. Difference of squares two expressions is equal to the product of the difference of these expressions and their sum.
a 2 - b 2 = (a -b) (a+b)
4. Cube of sum two expressions is equal to the cube of the first expression plus triple the product of the square of the first expression and the second plus triple 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 triple the product of the square of the first expression and the second plus triple 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 and 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 obtain
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 of the squares of two expressions, we get
Example 3.
Simplify an expression
(x - y) 2 + (x + y) 2
Let's 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)
