Algebra: Quadratic Functions and Factoring

• Algebra encompasses fundamental operations, factoring techniques, and progressions.

Quadratic Functions

• A quadratic function is a polynomial function of degree two.
• The general form is f(x) = ax² + bx + c, where a ≠ 0.
• The graph of a quadratic function is a parabola.
• The vertex of the parabola represents the maximum or minimum value of the function.
• The x-coordinate of the vertex is given by -b/2a.
• The discriminant, Δ = b² - 4ac, determines the nature of the roots:
  • Δ > 0: Two distinct real roots.
  • Δ = 0: One real root (a repeated root).
  • Δ < 0: No real roots (two complex roots).
• Quadratic equations can be solved by:
  • Factoring
  • Completing the square
  • Using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Factoring Techniques

• Factoring is the process of breaking down a polynomial into simpler expressions (factors) that, when multiplied together, give the original polynomial.
• Common factoring techniques include:
  • Greatest Common Factor (GCF): Identifying and factoring out the largest factor common to all terms.
  • Difference of Squares: a² - b² = (a + b)(a - b)
  • Perfect Square Trinomials: a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²
  • Factoring by Grouping: Grouping terms and factoring out common factors from each group.
  • Trinomial Factoring: Factoring quadratic trinomials of the form ax² + bx + c.
  • Sum/Difference of Cubes: Factoring expressions like a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²)

Binomial Theorem

• The binomial theorem describes the algebraic expansion of powers of a binomial.
• It states that (a + b)ⁿ = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n.
• (n choose k) represents the binomial coefficient, calculated as n! / (k!(n-k)!).
• n! (n factorial) is the product of all positive integers up to n.
• The binomial coefficients can be found using Pascal's Triangle.
• The theorem is used to expand expressions of the form (a + b)ⁿ without direct multiplication.

Geometric Progression (GP)

• A geometric progression is a sequence where each term is multiplied by a constant (common ratio) to get the next term.
• The general form is a, ar, ar², ar³,..., where 'a' is the first term and 'r' is the common ratio.
• The nth term of a GP is given by ar^(n-1).
• The sum of the first n terms of a GP is given by:
  • S_n = a(1 - rⁿ) / (1 - r) if r ≠ 1
  • S_n = na if r = 1
• The sum of an infinite GP is given by S_∞ = a / (1 - r) if |r| < 1.
• Geometric mean between two numbers a and b is √(ab).

Harmonic Progression (HP)

• A harmonic progression is a sequence where the reciprocals of the terms form an arithmetic progression.
• If a, b, c are in HP, then 1/a, 1/b, 1/c are in AP.
• The nth term of a HP is the reciprocal of the nth term of the corresponding AP.
• The harmonic mean (H) between two numbers a and b is given by H = 2ab / (a + b).
• The relationship between Arithmetic Mean (A), Geometric Mean (G), and Harmonic Mean (H) for two numbers is G² = AH.

Fundamental Operations in Algebra

• The fundamental operations in algebra are addition, subtraction, multiplication, and division.
• These operations are performed on variables, constants, and expressions.
• Addition and subtraction involve combining like terms.
• Multiplication involves distributing terms.
• Division involves simplifying fractions and rational expressions.
• Order of operations (PEMDAS/BODMAS) is crucial: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Indices (Exponents)

• Indices (or exponents) represent the power to which a number or variable is raised.
• Laws of indices:
  • a^m * a^n = a^(m+n)
  • a^m / a^n = a^(m-n)
  • (a^m)^n = a^(mn)
  • (ab)^n = a^n * b^n
  • (a/b)^n = a^n / b^n
  • a^0 = 1 (where a ≠ 0)
  • a^(-n) = 1 / a^n
  • a^(m/n) = nth root of a^m
• These laws are used to simplify and manipulate expressions involving exponents.

Algorithms

• An algorithm is a step-by-step procedure for solving a problem or performing a calculation.
• In algebra, algorithms are used for various tasks, such as:
  • Solving equations (linear, quadratic, etc.)
  • Factoring polynomials
  • Simplifying expressions
  • Performing long division
• Algorithms should be well-defined, finite, and effective.
• Examples include:
  • The Euclidean algorithm for finding the greatest common divisor (GCD).
  • Algorithms for solving systems of linear equations (e.g., Gaussian elimination).
  • Algorithms for polynomial division.