Algebra Basics: Real Numbers and Polynomials

Real Numbers

• A real number is a value that can be represented on the number line
• Can be expressed in decimal form, which may terminate or repeat
• Includes:
  • Natural numbers (1, 2, 3, ...)
  • Whole numbers (0, 1, 2, ...)
  • Integers (...,-2, -1, 0, 1, 2, ...)
  • Rational numbers (fractions, e.g., 3/4, 22/7)
  • Irrational numbers (non-repeating, non-terminating decimals, e.g., π, e)

Polynomials

• A polynomial is an expression consisting of variables (e.g., x, y) and coefficients (numbers)
• Form: a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0
• Where:
  • a_n is the leading coefficient
  • x is the variable
  • n is the degree of the polynomial (highest power of x)
• Examples:
  • Monomials: 3x, 2y^2
  • Binomials: x + 2, 3x^2 - 4x
  • Trinomials: x^2 + 3x - 2, 2x^3 - 5x^2 + x

Operations with Polynomials

• Addition and subtraction:
  • Combine like terms (terms with the same variable and exponent)
• Multiplication:
  • Distribute each term in one polynomial to each term in the other
• Properties:
  • Commutative: a + b = b + a and a × b = b × a
  • Associative: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
  • Distributive: a × (b + c) = a × b + a × c