Mathematics Specialised Level 4 key terms

For a quadratic ax² + bx + c, the value b² − 4ac. Its sign shows whether the equation has two distinct real roots, one repeated real root, or no real roots.

Rewriting a quadratic as a perfect-square expression plus a constant, for example x² + 6x + 1 = (x + 3)² − 8. It reveals the vertex and helps solve equations.

A polynomial P(x) has factor (x − a) exactly when P(a) = 0. The result converts root-finding into factor-finding and supports polynomial division.

When a polynomial P(x) is divided by (x − a), the remainder equals P(a). It gives remainders without performing long division.

The process of dividing one polynomial by another to obtain a quotient and remainder, analogous to numerical long division. It underpins partial fractions and the factor theorem.

Expressing a single rational expression as a sum of simpler fractions with linear or quadratic denominators. This makes integration and series expansion more manageable.

A line that a curve approaches ever more closely as it extends, without necessarily meeting it. Vertical, horizontal and oblique asymptotes describe the long-term or near-singular behaviour of rational functions.

A function formed as one polynomial divided by another, defined wherever the denominator is nonzero. Its zeros, vertical asymptotes and end behaviour follow from the numerator and denominator.

The turning point of a parabola, where the quadratic attains its maximum or minimum value. Its coordinates can be read from completed-square form or found by differentiation.

A pair of equations expressing x and y separately in terms of a third variable, called the parameter. They describe curves such as circles and parabolas conveniently.

An equation relating x and y directly, without a parameter, describing a curve in the plane. Converting from parametric form usually involves eliminating the parameter.

A sequence in which each term differs from the previous one by a fixed constant, the common difference. Its nth term and sum have standard closed formulas.

A sequence in which each term is the previous term multiplied by a fixed nonzero constant, the common ratio. Convergence of its sum depends on the size of that ratio.

The constant multiplier linking consecutive terms of a geometric progression, found by dividing any term by the one before it. Its magnitude determines whether a geometric series converges.

The limiting value of a geometric series whose common ratio has magnitude less than one, equal to a/(1 − r) for first term a. Otherwise the sum diverges.

A technique for summing a series by writing each term as a difference of two consecutive values, so that most terms cancel. It is useful for series such as partial fractions.

The sigma symbol Σ compactly denotes a sum of terms indexed over a stated range. It clarifies the general term and limits of a series.

The instantaneous rate of change of a function, defined as the limit of a difference quotient. Geometrically it gives the gradient of the tangent to the curve.

A rule for differentiating a composite function: the derivative of f(g(x)) is f′(g(x)) multiplied by g′(x). It handles nested expressions such as powers of trigonometric functions.

Differentiating an equation relating x and y term by term with respect to x, treating y as a function of x. It applies when y cannot easily be isolated.

A point where the derivative of a function is zero, so the tangent is horizontal. The second derivative or a sign change classifies it as maximum, minimum or point of inflection.

The derivative of the derivative, measuring how the gradient itself changes. Its sign at a stationary point indicates a local maximum or minimum.

A technique that reverses the chain rule by changing the variable of integration to simplify an integral. The differential and limits must be transformed consistently.

A technique derived from the product rule, written ∫u dv = uv − ∫v du. It suits products such as a polynomial multiplied by an exponential or logarithm.

The integral of a function between two limits, representing the signed area under its curve. It is evaluated using an antiderivative and the fundamental theorem of calculus.

The region between a curve and the x-axis over an interval, computed by a definite integral. Areas below the axis count as negative unless absolute values are used.

An angular measure in which one radian subtends an arc equal in length to the radius. A full turn is 2π radians, and calculus formulas for trigonometric functions assume this unit.

The distance along a curved path. For a circle of radius r, an arc subtending angle θ radians has length rθ.

The area of the region bounded by two radii and an arc of a circle, equal to ½r²θ when θ is measured in radians. It extends to areas swept by curves in polar form.

An equation involving trigonometric functions that holds for all valid angles, such as sin²θ + cos²θ = 1. Identities are used to simplify expressions and solve equations.

An identity expressing a trigonometric function of twice an angle in terms of the original angle, for example sin 2θ = 2 sin θ cos θ. These formulas simplify integrals and equation solving.

A product of two vectors giving a scalar, equal to the product of their magnitudes and the cosine of the angle between them. It tests perpendicularity and finds angles.

A quantity with both magnitude and direction, often written as a column of components or in unit-vector notation. Vectors model displacement, velocity and forces.

A vector from a fixed origin to a point, specifying the point's location. Differences of position vectors give displacement vectors between points.

Two nonzero vectors whose scalar product is zero, meaning the angle between them is a right angle. This condition is used to find unknown components.

A scalar associated with a square matrix; for a 2×2 matrix it equals ad − bc. A nonzero determinant means the matrix is invertible.

The matrix that multiplies a given square matrix to give the identity. It exists only when the determinant is nonzero and is used to solve linear systems.

A rectangular array of numbers arranged in rows and columns, used to represent transformations or systems of equations. Multiplication follows row-by-column rules and is not generally commutative.

The polar description of a complex number: its distance from the origin and its angle from the positive real axis. Together they allow multiplication and division in polar form.

The number obtained by reversing the sign of the imaginary part, so a + bi becomes a − bi. Multiplying a complex number by its conjugate gives a real value.

A plane in which a complex number a + bi is plotted at coordinates (a, b). It gives geometric meaning to modulus, argument and addition of complex numbers.

A number of the form a + bi, where a and b are real and i² = −1. Complex numbers extend the real line and appear as roots of polynomial equations.

The distribution of the number of successes in a fixed number of independent trials, each with the same probability of success. It is defined by the trial count and success probability.

A continuous, symmetric bell-shaped distribution determined by its mean and standard deviation. Many natural measurements approximate it, and probabilities come from standardising values.

A measure of spread equal to the square root of the variance, expressed in the same units as the data. It quantifies typical distance from the mean.

The average of the squared deviations from the mean, measuring how spread out values are. Its square root is the standard deviation.

The probability-weighted average of a random variable's possible values. For repeated trials it predicts the long-run mean outcome.

A complete description assigning probabilities to each possible value of a random variable. Discrete distributions list values; continuous ones use a density function.

A proof that assumes the statement is false and derives an impossibility, showing the assumption must fail. It suits results such as the irrationality of the square root of two.

A proof technique for statements about all positive integers: establish a base case, assume the statement for some integer, then prove it for the next. This chain covers every integer from the base onward.

A single case that shows a general statement is false. Finding one disproves a conjecture without any further argument.

A statement believed true on the basis of evidence but not yet proved. Once proved, it becomes a theorem.