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Aptitude Topics

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables. They provide powerful shortcuts for expanding and factoring expressions without lengthy multiplication.

Fundamental Principles

Identity

An equation that is true for every value of the variable(s). Unlike equations, identities don't need to be solved — they hold universally.

Key Square Identities

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

Cubic Identities

(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).

Essential Formulation Tips

  • Memorize the 7 standard identities — they appear in almost every exam.
  • Recognize the pattern before expanding — save time.
  • Use (a+b)² - 2ab = a² + b² to find sum of squares when sum and product are known.
  • Use (a+b)³ - 3ab(a+b) = a³ + b³ for cube problems.

Shortcut Execution Techniques

  • a² + b² = (a+b)² - 2ab.
  • a³ + b³ = (a+b)³ - 3ab(a+b).
  • (a+b+c)² = a² + b² + c² + 2(ab + bc + ca).
  • a³ + b³ + c³ - 3abc = (a+b+c)(a² + b² + c² - ab - bc - ca).

Contextual Inquiries (FAQs)

Q: What is the difference between an identity and an equation?

A: An identity holds for all values of variables, while an equation holds only for specific values.

Q: How do I remember a³ + b³?

A: Think: (sum)(square of first - product + square of second): (a+b)(a²-ab+b²).

Q: When do I use (a+b+c)²?

A: When expanding the square of a trinomial or finding a²+b²+c² given a+b+c and ab+bc+ca.