Frequently used formulas in algebra

Formulas for factoring

a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²

a² - b² = (a + b)(a - b)

a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)

a³ + 3a²b + 3ab² + b³ = (a + b)³a³ - 3a²b + 3ab² - b³ = (a - b)³

aⁿ - bⁿ = (a - b)(a^{n - 1} + a^{n - 2}b + a^{n - 3}b²........ + a²b^{n - 3}+ ab^{n
- 2} + b^{n - 1})for n > 1
(Note that
a² - b² = (a + b)(a - b) is a special case of the formula where n = 2, and
a³ - b³ = (a - b)(a² + ab + b²) is a special case of the formula when n = 3.)
In a special case where b = 1, we have
aⁿ - 1 = (a - 1)(a^{n - 1} + a^{n - 2} + a^{n -
3}........ + a²+ a + 1)

aⁿ + bⁿ = (a + b)(a^{n - 1} - a^{n - 2}b + a^{n - 3}b²........ + a²b^{n - 3}- ab^{n
- 2} + b^{n - 1})
where n is an odd natural number and n > 1
(Note that
a³ + b³ = (a + b)(a² - ab + b²) is a special case of the formula when n = 3.)

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