Algebraic Expressions and Identities

• Basic Concepts

  • Variable: Symbols representing unknown values (e.g., $x, y, z$)
  • Constant: Fixed numerical values (e.g., $3, -5, \frac{1}{2}$)
  • Term: Product of constants and variables (e.g., $4x$, $-7xy$)
  • Factor: Elements multiplied together to form a term
  • Coefficient: Numerical factor of a term (e.g., $4$ in $4x$)

• Types of Expressions

  • Monomial: Expression with 1 term (e.g., $7x$, $-3ab$)
  • Binomial: Expression with 2 terms (e.g., $a + b$, $4x - 3$)
  • Trinomial: Expression with 3 terms (e.g., $x + y + z$)
  • Polynomial: Expression with one or more terms with non-negative integer exponents

• Like and Unlike Terms

  • Like Terms: Terms having the same algebraic factors (e.g., $2x$ and $-5x$)
  • Unlike Terms: Terms having different algebraic factors (e.g., $2x$ and $3y$)

• Operations on Expressions

  • Addition & Subtraction
    • Group and combine like terms
    • Keep unlike terms separate
    • Methods: Column Method or Horizontal Method
  • Multiplication
    • Monomial by Monomial: Multiply coefficients, add powers of matching variables
    • Monomial by Polynomial: Apply Distributive Law [$a(b + c) = ab + ac$]
    • Polynomial by Polynomial: Multiply each term of the first by every term of the second

• Standard Algebraic Identities

  • Identity I: $(a + b)^2 = a^2 + 2ab + b^2$
  • Identity II: $(a - b)^2 = a^2 - 2ab + b^2$
  • Identity III: $(a + b)(a - b) = a^2 - b^2$
  • Identity IV: $(x + a)(x + b) = x^2 + (a + b)x + ab$

• Applications of Identities

  • Numerical Calculations: Computing squares easily (e.g., $103^2 = (100 + 3)^2$)
  • Product Evaluation: Multiplying numbers quickly (e.g., $98 \times 102 = (100 - 2)(100 + 2)$)
  • Simplification: Expanding and simplifying complex algebraic expressions