Formulas of abbreviated multiplication

What are abbreviated multiplication formulas? What are examples of abbreviated multiplication formulas? How do I calculate the square of the sum?

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1. Formulas of abbreviated multiplication. What is this?

Abbreviated multiplication formulas are the common name for formulas that help you move between sum and product.

Formulas of abbreviated multiplication are one of the most important formulas used in mathematics. Due to their universality, the ability to use them is very important.

Formulas of abbreviated multiplication are used in the multiplication or exponentiation of algebraic expressions. They can also be used to solve equations, extract the roots of a polynomial, and transform formulas.

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2.Basic examples of abbreviated multiplication formulas

There are many abbreviated multiplication formulas. Here are the most important of them:

(a + b) ² = a² + 2ab + b² (a − b) ² = a² - 2ab + b² a² − b² = (a − b) (a + b) a³ − b³ = (a − b) (a² + ab + b²) a³ + b³ = (a + b) (a² − ab + b²) (a + b) ³ = a³ + 3a²b + 3ab² + b³ (a − b) ³ = a³ - 3a²b + 3ab² - b³

3. Formulas of abbreviated multiplication. The square of the sum

Here is the formula for the square of the sum:

(a + b) ² = a² + 2ab + b²

It is not necessary to use the shortened multiplication formula when computing the square of the sum of two numbers. For example (x + 2) ²

In this case, the following equation can be used:

(x + 2) ² = (x + 2) (x + 2) = x² + 2x + 2x + 4 = x² + 4x + 4

You can also use the formula for shortened multiplication:

(x + 2) ² = x² + 2⋅x⋅2 + 4 = x² + 4x + 4

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4. Formulas of abbreviated multiplication. The square of the difference

Here is the formula for the square of the difference:

(a − b) ² = a² - 2ab + b²

The above formula is used in the same way as the formula for the square of the sum of two numbers.

(x − 1) ² = x² - 2x + 1 (x − 2) ² = x² - 4x + 4 (x − 3) ² = x² - 6x + 9 (x − 6) ² = x² - 12x + 36

5. Formulas of abbreviated multiplication. The difference of squares

Here is the formula for the difference of the squares of two numbers:

a² - b² = (a − b) (a + b)

Equation examples:

x² - 22 = (x − 2) (x + 2) x² - 32 = (x − 3) (x + 3) x² - 52 = (x − 5) (x + 5)

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6. Formulas of abbreviated multiplication. The sum of the cubes

Here is the formula for the sum of cubes of two numbers:

a³ + b³ = (a + b) (a² − ab + b²)

Equation examples:

x³ + 33 = (x + 3) (x² − 3x + 32) x³ + 125 = x³ +53 = (x + 5) (x² − 5x + 25)

7. Formulas of abbreviated multiplication. Cubes difference

Here is the formula for the difference of cubes of two numbers:

a³ - b³ = (a − b) (a² + ab + b²)

Equation examples:

x³ - 8 = x³ −23 = (x − 2) (x² + 2x + 4) x³ - 125 = x³ - 53 = (x − 5) (x² + 5x + 25)

8. Formulas of abbreviated multiplication. Sum cube

Here is the formula for the cube of the sum of two numbers:

(a + b) ³ = a³ + 3a²b + 3ab² + b³

Equation examples: (x + 1) ³ = x³ + 3x2 + 3x + 1 (x + 2) ³ = x³ + 6x2 + 12x + 8

9. Formulas of abbreviated multiplication. Cube of difference

Here is the formula for the cube of the difference between two numbers:

(a − b) ³ = a3 - 3a²b + 3ab² - b³

Equation examples:

(x − 1) ³ = x³ - 3x2 + 3x - 1 (x − 3) ³ = x³ - 9x2 + 27x - 27

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10. Formulas of abbreviated multiplication. The square of the sum of three expressions

Here is the formula for the square of the sum of three expressions:

(a + b + c) ² = a² + b² + c² + 2ab + 2ac + 2bc

Equation examples:

(1+2+3)2 = 1 + 4 + 9 + 4 + 8 + 12 = 38

These formulas also have versions for more components, e.g. for three:

(a + b-c) ² = a² + b² + c² + 2ab - 2ac - 2bc (a-b + c) ² = a² + b² + c² - 2ab + 2ac - 2bc (a-b-c) ² = a² + b² + c² - 2ab - 2ac + 2bc

In general, this formula can be applied to the square of any number of components. The differences should be presented as the sum of the terms of the opposite sign.

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