Algebra and Polynomials
Algebra and Polynomials
The Algebra and Polynomials strand of the MATSEC Intermediate Pure Mathematics paper tests manipulation before theory: index laws, surds, quadratic theory, the factor and remainder theorems, partial fractions, inequalities and the binomial expansion for positive integer n. Questions are usually structured, with an early part feeding a later one, so a sign slip early can cost three parts.
Core rules
- Remainder theorem: when p(x) is divided by (x − a) the remainder is p(a); (x − a) is a factor exactly when p(a) = 0. For a divisor (ax − b), evaluate p(b/a).
- Quadratic: the discriminant b² − 4ac decides two, one or no real roots; completing the square gives the vertex and the range.
- Partial fractions: improper fractions must be divided first; a repeated linear factor (x − a)² needs A/(x − a) + B/(x − a)².
- Binomial: (a + b)^n = Σ C(n,r) a^(n−r) b^r; the general term targets a named coefficient.
Worked example
Find the remainder when p(x) = 2x³ − 5x² + x − 4 is divided by (2x + 1). Set 2x + 1 = 0, so x = −1/2. Then p(−1/2) = 2(−1/8) − 5(1/4) + (−1/2) − 4 = −0.25 − 1.25 − 0.5 − 4 = −6. The remainder is −6. Do not divide by (x + 1/2) by mistake - the theorem uses the root of the actual divisor.
Traps
- Sign errors when substituting negative roots.
- Forgetting to divide improper rational expressions before splitting them.
- Writing x² > 4 as x > 2 only; the solution is x > 2 or x < −2.
- Losing the constant term when expanding a product of three brackets.