The remainder when (x^{3} – 2x + m) is divided by (x – 1) is equal to the remainder when (2x^{3} + x – m) is divided by (2x + 1). Find the value of m.
-
A.
(frac{-7}{8}) -
B.
(frac{-3}{8}) -
C.
(frac{1}{8}) -
D.
(frac{5}{8})
Correct Answer: Option C
Explanation
The remainder theorem states that if f(x) is divided by (x – a), the remainder is f(a).
(f(x) = x^{3} – 2x + m) divided by (x – 1), so that a = 1.
Remainder = (f(1) = 1^3 – 2(1) + m = -1 + m)
(f(x) = 2x^{3} + x – m) divided by (2x + 1), so that a = (frac{-1}{2})
(f(frac{-1}{2}) = 2(frac{-1}{2}^{3}) + (frac{-1}{2}) – m = frac{-3}{4} – m)
(implies m – 1 = frac{-3}{4} – m), collecting like terms,
(2m = frac{1}{4} therefore m = frac{1}{8})