Algebra II

A complex number extends the real numbers by introducing i, a formal square root of -1. Every complex number can be written a + bi with a, b real, and can be plotted as a point on a 2D plane (the Argand plane) — turning algebra involving square roots of negative numbers into geometry involving rotation and scaling.

The logarithm of a number is the exponent to which a fixed base must be raised to produce that number. If x = bʸ, then y is the logarithm of x to base b, written log_b x = y. For example, log₁₀ 1000 = 3 because 10³ = 1000. The logarithm is the inverse of exponentiation, and it turns multiplication into addition — the property that made it, for three centuries, the backbone of hand computation, and today the natural language of exponential growth, decibels, pH, entropy, and algorithmic complexity.

An exponential function has the form f(x) = a·bˣ, where the variable sits in the exponent rather than the base. Unlike polynomials, where growth is bounded by a fixed power, exponential functions grow (or decay) by a constant multiplicative factor over every equal step in x — this is what makes them the natural model for compound interest, population growth, radioactive decay, and viral spread.

An arithmetic sequence is a sequence in which each term differs from the previous one by a constant amount, called the common difference d. Because the increment is always the same, arithmetic sequences correspond to linear functions of the index n, and their sums (arithmetic series) have a clean closed-form formula discovered — according to legend — by a young Carl Friedrich Gauss.

A geometric sequence is a sequence in which each term is obtained from the previous one by multiplying by a constant ratio r. Because growth is multiplicative rather than additive, geometric sequences correspond to exponential functions of the index n. When |r| < 1, the corresponding infinite geometric series converges to a finite sum — a surprising and useful fact behind everything from repeating decimals to the economics of stimulus spending.

A sequence is an ordered list of numbers, a₁, a₂, a₃, ..., each indexed by a positive integer — formally, a function from the natural numbers to the reals. A series is the sum of the terms of a sequence, S = a₁ + a₂ + a₃ + .... Sequences describe discrete processes step by step (compound interest each period, populations each generation), while series answer the natural follow-up question: what do you get when you add all those steps together, and does that sum settle down to a finite value or grow without bound?

The Binomial Theorem gives a direct formula for expanding (x+y)ⁿ into a sum of terms, without multiplying out (x+y) by itself n times. Each term's coefficient is a binomial coefficient — the same numbers that appear in Pascal's Triangle and count combinations — making the theorem a bridge between algebra and combinatorics.

A polynomial function is a function of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, built from nonnegative integer powers of x combined with addition, subtraction, and constant multiplication. Polynomial functions are the smoothest, best-behaved functions in algebra — they're continuous and differentiable everywhere, have no asymptotes or holes, and are completely determined by a finite list of coefficients. Their graphs, roots, and end behavior are governed by the degree n and leading coefficient aₙ, making polynomials the natural next step after quadratics and the foundation for approximating almost any other function.

Polynomial long division divides one polynomial by another using the same step-by-step process as long division of numbers: divide, multiply, subtract, bring down, and repeat. It expresses any dividend f(x) as f(x) = d(x)·q(x) + r(x), where the remainder r(x) has degree strictly less than the divisor d(x) — the key tool for factoring higher-degree polynomials and finding rational roots.

The Rational Root Theorem narrows the search for rational roots of a polynomial with integer coefficients down to a short, finite list of candidates, built entirely from the polynomial's constant and leading coefficients. Instead of guessing blindly, you can test only p/q ratios where p divides the constant term and q divides the leading coefficient.

Synthetic division is a streamlined shorthand for dividing a polynomial by a linear factor (x - c), using only the coefficients arranged in a compact array. It produces the same quotient and remainder as polynomial long division but with far less writing, making it the standard tool for quickly testing candidate roots.

Function composition chains two functions together so the output of one becomes the input of the next: (f∘g)(x) = f(g(x)). It's the algebraic way of describing a multi-step process — apply g first, then feed the result into f — and it's the operation whose 'undo' defines inverse functions (f∘f⁻¹ = identity).

The inverse of a function f, written f⁻¹, is the function that undoes f: if f sends a to b, then f⁻¹ sends b back to a. Not every function has an inverse — only those that are one-to-one (injective), meaning no two different inputs ever produce the same output. Inverse functions are how we 'solve backwards': logarithms undo exponentials, square roots undo squaring (on the right domain), and arcsine undoes sine. Graphically, a function and its inverse are mirror images of each other across the line y = x.

A piecewise function is defined by different formulas on different parts of its domain, stitched together into a single function. Common real-world examples — tax brackets, shipping rates by weight, absolute value — all change their rule depending on which interval the input falls into.

Partial fraction decomposition reverses the process of combining fractions over a common denominator: it breaks a single complicated rational expression into a sum of simpler fractions, each with a lower-degree denominator. This is the key algebraic step that makes rational functions tractable for integration in calculus and for inverse Laplace/Z-transforms in engineering.

A rational function is a ratio of two polynomials, f(x) = p(x)/q(x), with q(x) not identically zero. Unlike polynomials, rational functions can have gaps (holes) and vertical/horizontal/slant asymptotes wherever the denominator vanishes or dominates, making their graphs qualitatively richer and requiring careful analysis of both numerator and denominator behavior.

A radical function contains its variable inside a root, most commonly a square root: f(x) = √(x). Radical functions arise as the inverses of power functions (√x undoes squaring on x≥0), and their domain and range are restricted by the requirement that expressions under an even root be nonnegative.

A system of equations in three variables (x, y, z) is solved by systematically eliminating variables until a single equation in one unknown remains, then back-substituting. Geometrically, each linear equation represents a plane in 3D space, and a unique solution corresponds to the single point where all three planes intersect.