Algebra

Important formula and identities

i. (a+b)² = a²+2ab+b² = (a-b)²+4ab

ii. (a-b)² = a²-2ab+b² = (a+b)²-4ab

iii. a²+b² = (a+b)²-2ab = (a-b)²+2ab

iv. a²-b² = (a+b)(a-b)

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

vi. a³+b³ = (a+b)³-3ab(a+b) = (a+b)(a²-ab+b²)

vii. a³-b³ = (a-b)³+3ab(a-b) = (a-b)(a²+ab+b²)

viii. a³+b³+c³-3abc = 1/2(a+b+c)[(a-b)²+(b-c)²+(c-a)²]

ix. a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)

x. (a+b)³ = a³+b³+3a²b+3ab² = a³+b³+3ab(a+b)

xi. a⁴-b⁴=(a²+b²)(a+b)(a-b)

xii. (a+b+c)² = a²+b²+c²+2ab+2bc+2ca

xiii. (a-b+c)² = a²+b²+c²-2ab-2bc+2ca

xiv. (a+b-c)² = a²+b²+c²+2ab-2bc-2ca

xv. (a-b-c)² = a²+b²+c²-2ab+2bc-2ca

xvi. (x-a)(x+b) = x²+(a+b)x+ab

xvii. (x-a)(x-b) = x²-(a+b)x+ab

xviii. a²(b+c)+b²(c-a)+c²(a+b)+2abc = (a+b)(b+c)(c+a)

xix. a²(b²-c²)-b²(c²-a²)+c²(a²-b²) = (a-b)(b-c)(c-a)

xx. ab(a-b)+bc(b-c)+ca(c-a) = -(a-b)(b-c)(c-a)

xi. (a+b)²+(a-b)²=2(a²+b²)

xii. (a+b)²-(a-b)² = 4ab

xiii. (a+b+c)³ = a³+b³+c³+3(a+b)(b+c)(c+a)

xiv. 2a²b²+2b²c²+2c²a²-a⁴-b⁴-c⁴ = (a+b+c)(a-b+c)(b+c-a)(a+b-c)

xv. (a+b+c)³-a³-b³-c³ = 3(a+b)(b+c)(c+a)