Polynomial roots: multiplicity and complex solutions
Optional topic: touching and crossing the axis, conjugate roots, the fundamental theorem of algebra and cubics.
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#algebra
Articles: 51
Complex numbers: a number with two coordinates
Optional topic: arithmetic of a+bi, conjugation, modulus, argument and multiplication as rotation and scaling.
Parameters: one formula, different solution counts
Optional topic: two hypothetical tariffs, a zero coefficient, feasible usage and a family of parabolas.
Mathematical induction: proving every step
Optional topic: base case, hypothesis and step in an odd-number sum proof, plus an invalid step.
Center and spread: what ten minutes conceal
Two datasets with the same mean: median, mode, range, variance and standard deviation.
Samples and charts: whom the data represent
Population, sampling, frequencies and distorted scales using five commute responses.
Random variables: why expectation is not a promise
A score distribution, weighted average and linearity of expectation using four cards.
Conditional probability: what known information changes
Two draws with and without replacement, a tree, a 2×2 table, independence and mutual exclusion.
Probability: a model and a frequency differ
Equally likely outcomes, events, axioms, frequency, the law of large numbers and geometric probability.
Counting: a team or different roles
Sum and product rules, permutations and combinations; repetition and the binomial theorem in a branch.
Integrals: adding small contributions
Changing water flow, rectangle sums, the fundamental theorem, signs, areas and volumes.
Antiderivatives: from a rate to a quantity
Why antiderivatives form a family, how an initial value selects one and how to check integrals.
Finding a maximum: derivatives and constraints
Area with fixed perimeter: critical points, endpoints, second derivatives and theorem conditions.
Derivative rules: follow every dependency
Sum, product, quotient and chain rules; derivatives with domains and units.
The derivative: change at one instant
From average velocity and secants to a limit, instantaneous velocity and tangents.
Limits, continuity and asymptotes
Why a limit may exist without a value at the point: sequences, discontinuities and infinity.
Trigonometric inequalities: when a signal exceeds a threshold
Circle arcs, time intervals, strict boundaries, periodic extension, a system and substitution.
Find the angle: inverse functions and trigonometric equations
Principal angles, all periodic solutions, time windows, substitution and factoring.
Trigonometric identities: angles and waves
Derive angle addition by rotating coordinates, then double angles and wave transformations.
Sinusoids: describing repeated change
Period, amplitude and phase, plus sin, cos, tan and cot graphs with their breaks.
From angles to numbers: radians and the unit circle
Rotation, arc divided by radius, and four trigonometric ratios.
Exponential and logarithmic inequalities
How the base controls comparison direction and the domain controls the final interval.
Exponential and logarithmic equations
Time to a threshold, substitution, logarithm domains and candidate checks.
Logarithms: which exponent is needed
Invert powers, derive logarithm rules, change bases and read a positive logarithmic axis.
Exponentials: equal factors instead of equal increments
Doubling, real exponents and e: read an exponential model and its limits.
Geometric progressions and compound percentages
Constant ratios, geometric sums and repeated percentages: distinguish a changing base from a fixed increment.
Sequences and arithmetic progressions
How an index identifies a term, why the formula uses n−1 and where the sum comes from: a hypothetical eight-week contribution schedule.
Power functions: why scale changes the result
Linear, square and cubic scaling, graphs of powers and roots, and negative and rational exponents with explicit domains.
How to read and transform an unfamiliar graph
Growth, symmetry, bounds and transformations, plus composition order and the condition for an inverse function: a resource graph and a reverse rule.
Parabolas: reading a quadratic relationship
Vertex, zeros and parabola transformations through the area of a rectangle with fixed perimeter: exact coordinates and model limits.
Inverse proportion: why doubling gives half
A constant product, the k/x graph and asymptotes: time for fixed work and equal-area rectangles, distinguished from an inverse function.
Linear functions: fixed fees and constant rates of change
Read the coefficients in y = kx + b, compare two tariffs and find their equal-price point with the correct units and model domain.
Functions: how input and output are connected
One rental tariff as a formula, table and graph: understand inputs, domain and range, and distinguish a rule from an individual calculation.
Absolute value: distance, tolerance and cases
Why a 50 ± 0.2 mm tolerance uses absolute value, how signs determine cases, and how an inner interval differs from two outer rays.
Sign charts: where an expression is positive or negative
Zeros, poles and multiplicity: build a sign table, include the correct endpoints and preserve an isolated solution.
Inequalities: describe every admissible value
Negative multipliers, open endpoints, intersections and unions: a budget example and two short proofs explain how to describe an entire solution set.
Rational and radical equations: why we check the answer
Denominators, square-root signs and nonlinear systems: distinguish an allowed candidate from a genuine solution of the original problem.
Reducing a harder equation to a simpler one
Substitution, biquadratic equations, polynomial division and the remainder theorem: reduce the degree and recover every allowed root.
Quadratic equations: the discriminant and Vieta’s formulas
Derive the quadratic formula by completing the square, find a plot’s dimensions and connect the discriminant, graphs and factors.
Two unknowns: when we need a system of equations
Recover two prices from two receipts: substitution, elimination and graphs explain one solution, none or infinitely many.
Turning a real-world problem into an equation
Joint work, motion and mixtures: choose an unknown, align units, write a relationship and check what the answer means.
Equations as balance: find the unknown without losing solutions
A bill with an unknown price explains equivalent transformations, why division can lose solutions and why squaring can introduce extraneous ones.
Roots and fractional powers: reversing multiplication
Recover a side from an area and an edge from a volume, explain why √(x²) is an absolute value, and check the conditions for fractional powers.
Rational expressions: cancellation keeps excluded values excluded
Cancel factors, find common denominators and preserve the original domain: a simplified fraction still has a hole.
Factoring: seeing the structure of an expression
Use common factors, grouping and identities to turn sums into products, reveal zeros and check a factorization.
Special products: where the middle term comes from
Derive squares of sums and differences, differences of squares and cube identities: why 2ab matters and how to calculate 99².
How monomials and polynomials work
From four parts of a rectangle to multiplying binomials: coefficients, degree, like terms, and adding and subtracting polynomials.
Powers and scientific notation for large and small numbers
Read integer exponents, understand why 10⁰ = 1 and 10⁻³ = 0.001, and compare large numbers without confusing GB with GiB.
Why replace numbers with letters?
One formula for different purchases: variables, parameters, substitution, identities, the distributive law, units and allowed values.
Fractions, proportions and percentages without confusion
Scale a recipe and see why discounts of 20% and 10% total 28%: fractions, proportions, comparison bases and percentage points.
Which numbers do measurements and calculations need?
From counting objects to a length of √2: number sets, rational and irrational numbers, and the difference between an exact value and rounding.