The standard algebraic identities are: (a+b)² = a²+2ab+b², (a−b)² = a²−2ab+b², and (a+b)(a−b) = a²−b². These identities are true for all values of a and b and are used to expand, simplify, and factorise algebraic expressions.
(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
(a+b)(a−b) = a² − b²
(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)
If a+b+c = 0, then a³+b³+c³ = 3abc
(a+b+c)² = a²+b²+c²+2(ab+bc+ca)
(a + b)² = a² + 2ab + b² Example: (x+3)² = x² + 6x + 9
(a − b)² = a² − 2ab + b² Example: (x−4)² = x² − 8x + 16
(a + b)(a − b) = a² − b² Example: (x+5)(x−5) = x² − 25
(x + a)(x + b) = x² + (a+b)x + ab Example: (x+2)(x+3) = x² + 5x + 6
(a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a+b) Example: (x+2)³ = x³ + 6x² + 12x + 8
(a − b)³ = a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a−b) Example: (x−1)³ = x³ − 3x² + 3x − 1
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
a³ + b³ + c³ − 3abc = (a+b+c)(a²+b²+c²−ab−bc−ca) Special case: if a+b+c = 0, then a³+b³+c³ = 3abc
(a+b)² + (a−b)² = 2(a²+b²)
(a+b)² − (a−b)² = 4ab
These are important for competitive exams.
Identity 1 for quick calculation: 99² = (100−1)² = 10000 − 200 + 1 = 9801
Identity 3 for quick calculation: 103 × 97 = (100+3)(100−3) = 100² − 9 = 10000 − 9 = 9991
Identity 7 for factorisation: 8x³ + 27 = (2x)³ + 3³ = (2x+3)(4x²−6x+9)
These shortcuts save time in competitive exams.
(a+b)² = a² + 2ab + b². For example, (x+5)² = x² + 10x + 25.
(a+b)(a−b) = a² − b². This is the difference of squares identity. Example: (x+7)(x−7) = x² − 49.
(a+b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a+b).
Example: Calculate 98². Use (100−2)² = 10000 − 400 + 4 = 9604. Or 52 × 48 = (50+2)(50−2) = 2500 − 4 = 2496.
When a+b+c = 0, then a³+b³+c³ = 3abc. This is a special case of the identity a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca).
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