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.
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,
\(2m = \frac{1}{4} \therefore m = \frac{1}{8}\)
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