Algebraic identities are the backbone of algebra. These formulas allow you to factorize, expand, and simplify expressions quickly without multiplication.
a²-b² = (a+b)(a-b).
(a+b)² = a² + 2ab + b².
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.
1. a² - b² = (a+b)(a-b) — Difference of Squares Example: x² - 9 = (x+3)(x-3)
2. (a+b)² = a² + 2ab + b² 3. (a-b)² = a² - 2ab + b² 4. (a+b)³ = a³ + 3a²b + 3ab² + b³ 5. (a-b)³ = a³ - 3a²b + 3ab² - b³
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
Example: If a=1, b=2, c=-3 → a+b+c = 0 ∴ 1³ + 2³ + (-3)³ = 3(1)(2)(-3) 1 + 8 - 27 = -18 ✓
Find the value of 103² - 97² Using a²-b² = (a+b)(a-b): = (103+97)(103-97) = 200 × 6 = 1200
a² - b² = (a + b)(a - b). This is called the 'difference of two squares' identity.
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca).
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