Algebra and Complex Numbers
The Algebra and Complex Numbers domain of GCE A/L Combined Mathematics tests whether you can manipulate polynomial and rational expressions fluently, reason about roots without solving, and switch between the Cartesian and polar descriptions of a complex number. Questions frequently chain sub-topics in one part: the factor theorem produces a quadratic whose roots then need a modulus–argument treatment.
Core rules
- For ax^2 + bx + c = 0: sum of roots = -b/a, product = c/a; the discriminant D = b^2 - 4ac decides distinct real, equal or complex roots.
- Remainder theorem: the remainder when p(x) is divided by (x - a) is p(a), and (x - a) is a factor exactly when p(a) = 0. For (ax - b) evaluate p(b/a).
- Binomial theorem: (1 + x)^n = 1 + nx + n(n-1)x^2/2! + ... valid for |x| < 1 when n is not a positive integer.
- Complex numbers: z = x + iy = r(cos θ + i sin θ) with r = |z| = sqrt(x^2 + y^2) and tan θ = y/x, the quadrant fixed by the signs of x and y. The principal argument lies in (-π, π].
- De Moivre: (r cis θ)^n = r^n cis(nθ); the nth roots are r^(1/n) cis((θ + 2kπ)/n) for k = 0, 1, ..., n - 1.
Worked example
Let z = -1 + i√3. Here x < 0 and y > 0, so the point is in the second quadrant: r = sqrt(1 + 3) = 2 and the acute angle is arctan(√3/1) = π/3, giving arg z = π - π/3 = 2π/3. Hence z = 2 cis(2π/3) and z^6 = 2^6 cis(4π) = 64 cis 0 = 64.
Traps
- Writing arg z = arctan(y/x) without correcting the quadrant; the negative real part is the whole point of the example above.
- Confusing z^2 with |z|^2, which are equal only when z is real.
- Multiplying or dividing an inequality by an expression of unknown sign, silently reversing its direction.
- Losing the repeated-root case when D = 0, or dropping the constant of a partial fraction with a repeated linear factor.