When f(x) is divided by x+2 and x-4, the remainders are 6 and 3 respectively. Find the remainder when f(x) is divided by x^2-2x-8.?

f(x) = q(x)(x² - 2x - 8) + r(x)
f(x) = q(x)(x + 2)(x - 4) + r(x)

The divisor is quadratic, so the remainder must have degree no greater than one. Functions q(x) and r(x) are both polynomials.

Let r(x) = Ax + B.

f(-2) = 6
q(-2)(-2 + 2)(-2 - 4) + A(-2) + B = 6
-2A + B = 6

f(4) = 3
q(4)(4 + 2)(4 - 4) + A(4) + B = 3
4A + B = 3

-2A + B = 6
4A + B = 3 (-)
----------------
-6A = 3
A = -1/2

4A + B = 3
4(-1/2) + B = 3
B = 5

r(x) = (-1/2)x + 5

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