Formulas or rules of abbreviated multiplication are used in arithmetic, or rather, in algebra, for a faster process of calculating large algebraic expressions. The formulas themselves are obtained from the rules existing in algebra for the multiplication of several polynomials.
Using 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 get the expression on the left side of the equality 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, since they are often used in solving problems and equations. The main formulas included in this list and their name are listed below.
Amount squared
To calculate the square of the sum, it is necessary to find the sum consisting of the square of the first term, the double 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 + s) ² = a² + 2as + s².
Difference square
To calculate the square of the difference, it is necessary to calculate the sum consisting of the square of the first number, the double 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 is as follows: (a - c) ² = a² - 2ac + s².
Square difference
The formula for the difference of two squared numbers is the product of the sum of these numbers by their difference. In the form of an expression, this rule looks as follows: a² - s² = (a + s) · (a - s).
Cube Amount
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, the 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 is as follows: (a + s) ³ = a³ + 3a²s + 3as² + s³.
Sum of cubes
According to the formula, the sum of the cubes is equated to the product of the sum of the given terms by their incomplete square of the difference. In the form of an expression, this rule is as follows: a³ + c³ = (a + c) · (a² - ac + s²).
Example. It is necessary to calculate the volume of the figure, which is formed by adding two cubes. Only the values of their sides are known.
If the values of the sides are small, then the calculations are simple.
If the lengths of the sides are expressed in bulky numbers, then in this case it is easier to apply the formula "Sum of cubes", which will greatly simplify the calculation.
Cube of difference
The expression for the cubic difference is as follows: as the sum of the third power of the first term, the tripled negative product of the square of the first term and the second, the triple product of the first term and the square of the second and negative cube of the second term. In the form of a mathematical expression, the difference cube is as follows: (a - c) ³ = a³ - 3a²s + 3as² - s³.
Cubes difference
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 of a mathematical expression, the difference of the cubes is as follows: a 3 - c 3 = (a - c) (a 2 + ac + c 2 ).
Example. It is necessary to calculate the volume of the figure, which will remain after subtracting from the volume of the blue cube the yellow volume figure, 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 pretty simple. And if the lengths of the sides are expressed in significant numbers, then you should apply the formula entitled "Difference of cubes" (or "Difference Cube"), which greatly simplifies the calculation.