Difference of squares
We derive the formula for the difference of squares $a^2-b^2$.
To do this, remember the following rule:
If any monomial is added to the expression and the same monomial is subtracted, then we get the correct identity.
Let's add to our expression and subtract from it the monomial $ab$:
In total, we get:
That is, the difference of the squares of two monomials is equal to the product of their difference and their sum.
Example 1
Express as a product of $(4x)^2-y^2$
\[(4x)^2-y^2=((2x))^2-y^2\]
\[((2x))^2-y^2=\left(2x-y\right)(2x+y)\]
We derive the formula for the sum of cubes $a^3+b^3$.
Let's take the common factors out of brackets:
Let's take $\left(a+b\right)$ out of brackets:
In total, we get:
That is, the sum of the cubes of two monomials is equal to the product of their sum by incomplete square their differences.
Example 2
Express as a product $(8x)^3+y^3$
This expression can be rewritten in the following form:
\[(8x)^3+y^3=((2x))^3+y^3\]
Using the difference of squares formula, we get:
\[((2x))^3+y^3=\left(2x+y\right)(4x^2-2xy+y^2)\]
We derive the formula for the difference of cubes $a^3-b^3$.
To do this, we will use the same rule as above.
Let's add to our expression and subtract from it the monomials $a^2b\ and\ (ab)^2$:
Let's take the common factors out of brackets:
Let's take $\left(a-b\right)$ out of brackets:
In total, we get:
That is, the difference of the cubes of two monomials is equal to the product of their difference by the incomplete square of their sum.
Example 3
Express as a product of $(8x)^3-y^3$
This expression can be rewritten in the following form:
\[(8x)^3-y^3=((2x))^3-y^3\]
Using the difference of squares formula, we get:
\[((2x))^3-y^3=\left(2x-y\right)(4x^2+2xy+y^2)\]
Example 4
Multiply.
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 difference of squares formula, 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 cubes of 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 cubes of 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 previous lessons, we considered two ways to factorize a polynomial: taking the common factor out of brackets and the grouping method.
In this lesson, we will look at another way to factorize a polynomial using abbreviated multiplication formulas.
We recommend that you write each formula at least 12 times. For better memorization, write down all the abbreviated multiplication formulas for yourself on a small cheat sheet.
Recall 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 of cubes is not very easy to remember, so we recommend using a special way 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 3Consider an example. It is necessary to factorize the difference of cubes.
Note that “27a 3" is "(3a) 3", which means that for the formula for the difference of cubes, instead of "a", we use "3a".
We use the formula for the difference of cubes. In place of “a 3”, we have “27a 3”, and in place of “b 3”, as in the formula, we have “b 3”.
Let's consider another example. It is required to convert the product of polynomials to 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 formula for the difference of cubes "", only instead of " a" is " x", And in place of " b" is " 1" .
For “(x − 1)(x 2 + x + 1)”, we use the formula for the difference of cubes in the opposite direction.
Let's consider a more difficult 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 formula for the difference of cubes
« a 3 − b 3 = (a − b)(a 2 + ab + b 2)”, then we can understand that in place of “ a" from the first bracket is " y 2, and in place of " b" is " 1".
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 prompt solution of various math 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, it is necessary 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, 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 volume of the blue cube from the volume of the figure yellow color, which is also a 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 multiplication formulas.
Studying the formulas for abbreviated multiplication: 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; sums and differences of cubes of two expressions.
Application of abbreviated multiplication formulas when solving examples.
To simplify expressions, factorize polynomials, and reduce polynomials to a 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 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 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 \u003d (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 times the second plus three times the product of the first expression times 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 of two expressions is equal to the cube of the first expression minus 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 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 \u003d (a + b) (a 2 - ab + b 2)
7. Difference of cubes of 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 \u003d (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 squared difference of two expressions, we obtain
98 2 \u003d (100 - 2) 2 \u003d 100 2 - 2 100 2 + 2 2 \u003d 10000 - 400 + 4 \u003d 9604
Example 2
Calculate
Using the formula for the difference of the squares of two expressions, we obtain
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 \u003d x 2 - 2xy + y 2 + x 2 + 2xy + y 2 \u003d 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 \u003d (a + b) (a 2 - ab + b 2)
a 3 - b 3 \u003d (a - b) (a 2 + ab + b 2)
