Algebra

Algebra Fundamental Concepts

  • Functions: A relationship between sets of numbers where each input corresponds to exactly one output.
  • Equations: Mathematical statements that assert the equality of two expressions.
  • Inequalities: Expressions where one side is greater or smaller than the other side.

Algebraic Operations

  • Polynomials: The set comprising mathematical expressions involving variables and coefficients.
  • Factorising: The process of decomposing an algebraic expressions into its factors.
  • Expanding brackets: The application of the distributive law to eliminate parentheses in algebraic expressions.
  • Simplifying: The process of making an equation, formula, or expression more manageable by eliminating unnecessary parts or by substitifying equivalent entities.

Quadratic Equations

  • Completing the square: A technique to convert a quadratic equation from standard form to vertex form.
  • Quadratic formula: A formula providing a solution to the quadratic equation.
  • Discriminant: A quantity that can determine the nature of roots of the quadratic equation.

Algebraic Expressions

  • Rational Expressions: An expression in the form of the ratio of two polynomials.
  • Surds: Any root sign expression. They are irrational numbers and hence cannot be simplified to a rational number.
  • Logarithms: An exponent to which a specified base must be raised to obtain a given number.

Sequences and Series

  • Arithmetic series: Sequences in which each term after the first is obtained from the preceding term by adding a constant difference.
  • Geometric series: Sequences in which each term after the first is obtained from the preceding term by multiplying by a fixed, non-zero number called the common ratio.
  • Summation: The process of adding sequences based on the sigma notation.

Progressions

  • Arithmetic Progressions (AP): A sequence of numbers in which the difference of any two successive members is a constant.
  • Geometric Progressions (GP): A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • Working with series in AP and GP: Understand how to evaluate the sum to n terms in AP and GP, and the sum to infinity in a GP.

Binomial Theorem

  • Binomial expansion: The process of expanding expressions that are raised to a particular power using the binomial theorem.
  • Coefficient of a term in binomial expansion: Understand how to find the coefficient of a particular term in a binomial expansion.
  • Binomial theorem: A fundamental theorem in algebra that describes the algebraic expansion of powers of a binomial.