Algebraic identities

(1) (a + b)² = a² + 2 ab + b² (2) (a + b)² = (a - b)² + 4 ab
(3) (a - b)² = a² - 2 ab + b² (4) (a - b)² = (a + b)² - 4 ab
(5) a² - b² = (a + b) (a - b)
(6) (x+a) (x+b) =x² + (a + b) x+a b
(7) (a+b)³=a³+3a²b+3ab²+b³ (8) (a+b)³=a³+b³+3ab(a+b)
(9) (a-b)³=a³-3a²b+3ab²-b³ (10) (a-b)³=a³-b³-3ab(a-b)
(11)  a³+b³ = (a+b)(a²-ab+b²) (12)  a³+b³=(a+b)³-3 ab(a + b)
(13) a³-b³= (a-b)(a²+ab+ b²) (14) a³-b³=(a-b)³ +3ab(a-b)

(15) (a+b+c)²= a²+b²+c² +2ab+2bc+2ca (16) (a+b-c)²=a²+b²+c² +2ab-2bc-2ca (17) (a-b+c)²= a²+b²+c²-2ab-2bc+2ca (18) (a-b-c)²= a²+b²+c²-2ab+2bc-2ca (19) a² + b² = (a + b)² - 2ab (20) a² + b² = (a - b)² + 2ab (21) a² +  b²=½ [(a+b)²-(a-b)²] (22) ab = ¼[(a+b)²- (a - b)²]

(23) (a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3b²c + 3ac² + 3bc² + 6abc

(24) (a + b - c)³ = a³ + b³ - c³ + 3a²b - 3a²c + 3ab² - 3b²c + 3ac² + 3bc² - 6abc

(25) (a - b + c)³ = a³ - b³ + c³ - 3a²b + 3a²c + 3ab² + 3b²c + 3ac² - 3bc² - 6abc

(26) (a - b - c)³ = a³ - b³ - c³ - 3a²b - 3a²c + 3ab² - 3b²c + 3ac² - 3bc² + 6abc

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

(28)(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴)

(29)(a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴)

(30)a4 – b⁴ = (a – b)(a + b)(a² + b²)

(31)a5 – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)

(32)if  a + b + c=0; then a³ + b³ + c³ =0

(33)a³ + b³ + c³- 3abc = (a + b+ c) (a² + b² + c² - ab - ac - bc)