ZIMSEC A Level · Pure Mathematics · Form 5
Quadratics
- Completing the square: ax² + bx + c = a(x + b/2a)² + c − b²/4a. The vertex is at (−b/2a, c − b²/4a).
- Discriminant b² − 4ac: > 0 two distinct real roots; = 0 equal roots; < 0 no real roots.
- A line meets a curve where their equations are solved simultaneously; a tangent gives a discriminant of zero.
Worked example: Find the values of k for which x² + kx + 9 = 0 has equal roots.
b² − 4ac = 0 → k² − 36 = 0 → k = ±6
Polynomials
- Remainder theorem: when p(x) is divided by (x − a), the remainder is p(a).
- Factor theorem: (x − a) is a factor if p(a) = 0.
Worked example: Show that (x − 2) is a factor of x³ − 3x² − 4x + 12 and factorise fully.
p(2) = 8 − 12 − 8 + 12 = 0, so (x − 2) is a factor.
Dividing: x³ − 3x² − 4x + 12 = (x − 2)(x² − x − 6) = (x − 2)(x − 3)(x + 2)
Partial fractions
- Distinct linear factors: A/(x − a) + B/(x − b).
- Repeated factor: A/(x − a) + B/(x − a)².
- Irreducible quadratic: (Bx + C)/(x² + k).
Worked example: Express (5x + 1)/((x + 1)(x − 1)) in partial fractions.
5x + 1 = A(x − 1) + B(x + 1). x = 1: 6 = 2B → B = 3. x = −1: −4 = −2A → A = 2.
Answer: 2/(x + 1) + 3/(x − 1)
Exam tip: Check partial fractions by recombining them or substituting a value of x.
Check yourself
- Find the remainder when x³ + 2x − 5 is divided by (x + 1).
- Write x² − 6x + 11 in completed square form and state the minimum value.
- For what values of m does y = mx − 1 not meet y = x² + 3?
Show answers
- p(−1) = −1 − 2 − 5 = −8
- (x − 3)² + 2; minimum value 2 at x = 3
- x² − mx + 4 = 0 has no real roots when m² − 16 < 0, i.e. −4 < m < 4
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