Cube of difference and difference of cubes: rules for applying formulas of reduced multiplication

Formulas or rules of abridged 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.

The use of these formulas provides a fairly prompt solution of various mathematical problems, and also helps to simplify expressions. Algebraic transformation rules allow you to perform some manipulations with expressions, following which you can get the expression on the right-hand side of the equation or convert the right-hand side of the equation (to get the expression on the left side after the equal sign).

It is convenient to know the formulas used for abridged multiplication, for memory, since they are often used in solving problems and equations. The main formulas included in this list and their name are listed below.

The square of the sum

To calculate the square of the sum, it is necessary to find a sum consisting of the square of the first summand, the doubled 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.

The square of the difference

To calculate the square of the difference, it is necessary to calculate the sum consisting of the square of the first number, the doubled 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 of two numbers, squared, is equal to the product of the sum of these numbers by their difference. As an expression, this rule looks like this: a² - с² = (a + с) · (a - с).

Cube amount

To calculate the cube of the sum of two summands, it is necessary to calculate the sum consisting of the cube of the first term, the tripled product of the square of the first summand and the second, the tripled product of the first summand and the second in the square, and the cube of the second summand. As an expression, this rule looks like this: (a + c) ³ = a³ + 3a²s + 3ac² + c³.

Sum of cubes

According to the formula, the sum of cubes is equated to the product of the sum of these terms by their incomplete square of the difference. As an expression, this rule looks like this: 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 dimensions 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 cumbersome numbers, then in this case it is easier to apply the formula "Sum of cubes", which will greatly simplify the calculations.

Cube 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 by the second, the tripled product of the first term by the square of the second and negative cube of the second term. In the form of a mathematical expression, the difference cube looks like this: (a - c) ³ = a³ - 3а²с + 3ас² - с³.

Difference of cubes

The difference formula for cubes differs from the sum of cubes with 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 in cubes looks like this: a ³ - with ³ = (a - c) (a ² + ac + c ² ).

Example. It is necessary to calculate the volume of the figure that will remain after subtracting from the volume of the blue cube a three-dimensional shape of 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, the calculations are fairly simple. And if the lengths of the sides are expressed in significant numbers, then it is worth applying the formula, entitled "Difference cubes" (or "Cube difference"), which greatly simplifies the calculation.