Question

Polynomial f(x) has a remainder of 3 when divided by x-2 and a remainder of -5 when it is divided by x+2. Determine the remainder when the polynomial is divided by x^2-4.

Answer 1

The remainder when f is divided by (x+2)(x−2) is at most a first degree polynomial. That means that f(x)= g(x)⋅(x+2)(x−2) + (ax+b) for some real numbers a, and b

f(x) = g(x) (x + 2)(x - 2) + ax + b.

We are told that when f(x) is divided by x-2 the remainder is 3 , so
f(2) = g(2) (4)(0) + (2a) + b
3 = 0 + 2a + b
2a + b = 3 **

when f(x) is divided by x+2 the remainder is -5
f(-2) = g(-2) (0)(-4) - 2a + b
-5 = 0 - 2a + b
2a - b = 5 ***
add ** and ***
4a = 8
a = 2 , then b = -1

so the remainder is 2x - 1