Abbreviated multiplication formulas: complete FSU table

📌 Abbreviated multiplication formulas (AMF) are the 7 basic formulas that help you quickly expand brackets and factor polynomials: square of a sum, square of a difference, difference of squares, cube of a sum, cube of a difference, sum of cubes, difference of cubes. They are used in algebra for solving equations and simplifying expressions.

Abbreviated multiplication formulas (AMF) are rules that allow you to quickly multiply polynomials, factor expressions, and simplify calculations. They are widely used in algebra, for solving equations, and in preparation for the EGE and OGE.

Abbreviated multiplication formulas are frequently encountered cases of polynomial multiplication; they are used for factoring polynomials, simplifying expressions, and reducing polynomials to standard form. All of them are proved by directly expanding the brackets and combining like terms.

Quick reference for abbreviated multiplication formulas

Formula	Result
(a + b)²	a² + 2ab + b²
(a − b)²	a² − 2ab + b²
a² − b²	(a − b)(a + b)
(a + b)³	a³ + 3a²b + 3ab² + b³
(a − b)³	a³ − 3a²b + 3ab² − b³
a³ + b³	(a + b)(a² − ab + b²)
a³ − b³	(a − b)(a² + ab + b²)

Formulas for squares (2nd power)
Formulas for cubes (3rd power)
Formulas for the 4th power
Formulas for the nth power
Examples of use

Formulas for squares (2nd power)

Square of a sum

{(a + b)}^{2} = a^{2} + 2 a b + b^{2}

Square of a difference

{(a - b)}^{2} = a^{2} - 2 a b + b^{2}

Difference of squares

a^{2} - b^{2} = (a - b) (a + b)

{(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 a c + 2 b c

Formulas for cubes (3rd power)

Cube of a sum

{(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

Cube of a difference

{(a - b)}^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

Sum of cubes

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

Difference of cubes

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

Formulas for the 4th power

{(a + b)}^{4} = a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}

{(a - b)}^{4} = a^{4} - 4 a^{3} b + 6 a^{2} b^{2} - 4 a b^{3} + b^{4}

a^{4} - b^{4} = (a - b) (a + b) (a^{2} + b^{2})

Formulas for the nth power

{(a + b)}^{n} = a^{n} + n a^{n - 1} b + n (n - 1) 2 a^{n - 2} b^{2} + ... + n! k! (n - k)! a^{n - k} b^{k} + ... + b^{n}

{(a - b)}^{n} = a^{n} - n a^{n - 1} b + n (n - 1) 2 a^{n - 2} b^{2} + ... + {(- 1)}^{k} n! k! (n - k)! a^{n - k} b^{k} + ... + {(- 1)}^{n} b^{n}

Examples of using abbreviated multiplication formulas

Example 1: Square of a sum

$(3 x + 2) 2 = (3 x) 2 + 2 \cdot 3 x \cdot 2 + 2^{2} = 9 x^{2} + 12 x + 4$

Example 2: Difference of squares

$25 x^{2} - 16 = (5 x) 2 - 4^{2} = (5 x - 4) (5 x + 4)$

Example 3: Cube of a sum

$(2 a + 1) 3 = (2 a) 3 + 3 \cdot (2 a) 2 \cdot 1 + 3 \cdot 2 a \cdot 1^{2} + 1^{3} = 8 a^{3} + 12 a^{2} + 6 a + 1$

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