MATSEC IM Pure Mathematics key terms

An expression made by adding terms of the form ax^n, where n is a non-negative whole number and a is a constant. Its degree is the highest power of x present.

If a polynomial f(x) gives f(a)=0, then (x−a) is a factor of f(x). Conversely, if (x−a) is a factor, then f(a)=0.

When a polynomial f(x) is divided by (x−a), the remainder equals f(a). This lets you find remainders without doing long division.

For the quadratic ax²+bx+c=0, the discriminant is b²−4ac. Its sign tells you whether the roots are real and distinct, real and equal, or not real.

Rewriting a quadratic ax²+bx+c in the form a(x+h)²+k. It reveals the vertex of the parabola and helps solve equations that do not factorise easily.

The expansion of (a+b)^n using binomial coefficients, giving terms a^n, a^(n−1)b, and so on. For a positive whole number n the expansion has n+1 terms.

Splitting a single rational expression into a sum of simpler fractions with linear or quadratic denominators. Useful before integrating rational functions or expanding them.

The set of all input values for which a function is defined. For example, the domain of 1/x excludes x=0 because division by zero is undefined.

The set of all output values a function actually produces as its input varies over the domain. It is found from the graph or by analysing the rule.

A function formed by applying one function and then another, written fg(x) meaning f(g(x)). The output of g becomes the input of f.

The function that reverses the effect of a one-to-one function f, written f⁻¹. It satisfies f⁻¹(f(x))=x, and its graph is the reflection of f in y=x.

A straight line that a curve approaches more and more closely as x or y tends to infinity, without normally meeting it. Graphs of rational and exponential functions often have them.

A function of the form f(x)=a^x, where a is a positive constant not equal to 1. Its graph always passes through (0,1) and has the x-axis as an asymptote.

For base a and number N, the logarithm log_a N is the power p such that a^p=N. Logarithms turn multiplication into addition through their laws.

The logarithm to base e, written ln x. It is the inverse of the exponential function e^x, and its derivative is 1/x for x>0.

A sequence in which each term after the first is obtained by adding a fixed constant, the common difference, to the previous term. The nth term is a+(n−1)d.

A sequence in which each term after the first is obtained by multiplying the previous term by a fixed constant, the common ratio. The nth term is ar^(n−1).

The value that the sum of the first n terms of a convergent geometric progression approaches as n grows without limit. It exists only when the common ratio satisfies |r|<1, and equals a/(1−r).

A compact way of writing a sum using the Greek letter Σ. The expression below and above the sigma gives the starting and finishing values of the index.

A unit of angle defined so that one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. π radians equal 180 degrees.

In any triangle, a/sin A = b/sin B = c/sin C, where the sides are opposite the corresponding angles. It is used when you know a matching side and angle.

In any triangle, a² = b² + c² − 2bc cos A. It relates all three sides to one angle and is used when the sine rule cannot be applied directly.

An identity expressing a trigonometric ratio of a sum or difference of two angles, such as sin(A+B)=sin A cos B + cos A sin B. These are used to simplify and solve trigonometric expressions.

Identities giving trigonometric ratios of 2A in terms of ratios of A, for example sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A.

A measure of how steep a straight line is, calculated as the change in y divided by the change in x between two points on the line. Parallel lines have equal gradients.

The equation (x−a)²+(y−b)²=r² represents a circle with centre (a,b) and radius r. Expanding it gives the general form x²+y²+2gx+2fy+c=0.

For vectors a and b, the scalar product is a·b = |a||b| cos θ, where θ is the angle between them. It gives a number, not a vector, and equals zero for perpendicular vectors.

The vector that locates a point relative to a fixed origin, usually written OA or r. It gives both the direction and distance of the point from the origin.

A vector with magnitude 1. Any non-zero vector a has a unit vector in its direction given by a divided by |a|.

For z = x + iy, the modulus is |z| = √(x²+y²). It is the distance of the point representing z from the origin on an Argand diagram.

The angle that the line from the origin to the point representing z makes with the positive real axis, measured anticlockwise. It is usually given in radians.

For z = x + iy, the conjugate is z̄ = x − iy, found by changing the sign of the imaginary part. Multiplying a complex number by its conjugate gives a real number.

Writing a complex number as r(cos θ + i sin θ), where r is the modulus and θ the argument. It is convenient for multiplication, division and powers of complex numbers.

The rate at which a function changes with respect to its variable, written dy/dx or f′(x). Geometrically it is the gradient of the tangent to the curve at a point.

A rule for differentiating a function of a function: if y = f(u) and u = g(x), then dy/dx = dy/du × du/dx. It is essential for composite expressions.

A point on a curve where the gradient is zero, so dy/dx = 0. It may be a maximum, a minimum, or a point of inflection.

The integral of a function between two limits, written ∫ from a to b of f(x) dx. It gives the signed area between the curve and the x-axis.

A technique that reverses the chain rule: a new variable u replaces part of the integrand, and dx is replaced using du = (du/dx) dx, simplifying the integral.

A numerical method for estimating a definite integral by dividing the area under a curve into trapezia and adding their areas. Accuracy improves as the number of strips increases.

The volume formed when a region under a curve is rotated through one full turn about an axis. It is found by integrating πy² with respect to x, between appropriate limits.