1. Remainder and factor theorem revision and challenging questions O-level Additional Maths

a. State the Remainder Theorem.
b. The polynomial f(x) leaves a remainder 5 when divided by (x-1) and remainder -1 when divided by (x+2). Find the remainder when f(x) is divided by (x-1)(x+2).

Comments:
Students are expected to be familiar with Remainder theorem and Division algorithm for polynomials. Most students are familiar with Remainder theorem, but not Division algorithm. To solve part b students need to have knowledge of Divison Algorithm.

Answers:
a. When a polynomial f(x) is divided by the linear polynomial ax + b, its remainder can be expressed as
f(-b/a).

b. f(1) = 5, f(-2) = -1.

When f(x) is divided by (x-1)(x+2), the remainder should be linear, ie. of the form ax + b.

Using the division algorithm for polynomials,

Dividend = Divisor x Quotient + Remainder,

f(x) = (x - 1)(x + 2) x Q(x) + ax + b
f(1) = a + b, and f(-2) = -2a + b
5 = a + b, -1 = -2a + b,

giving a = 2 and b = 3.

Hence the remainder is 2x + 3.