Question

What is the remainder when (x4 + 36) is divided by (x2 – 8)?
36
64
100
117

Answer 1

Let the remainder be ax + b (degree < 2). For x^2 = 8 (i.e. x = ±2√2) we have

x^4 + 36 = ax + b.

But x^4 = (x^2)^2 = 8^2 = 64, so 64 + 36 = 100 = ax + b for x = ±2√2. Thus
100 = 2√2 a + b and 100 = −2√2 a + b, which imply a = 0 and b = 100.

Remainder = 100.