If x + 2/x = 3 what is the value of x^3 + 8/x^3 = ?

Problems like these are the reason why I keep emphasizing on remembering the Algebraic identities. If you know algebraic identities, applying the appropriate one and solving this question will take you less than 30 seconds.

The algebraic identity to be used in this question is \((a+b)^3\) = \(a^3\) + \(b^3\) + 3ab(a+b).

In the problem given, x + \(\frac{2}{x}\) = 3. We can take a = x and b = \(\frac{2}{x}\); when we do this, we can quickly see that we are trying to find \(a^3\) + \(b^3\).
From the identity above, we see that \(a^3\) + \(b^3\) = \((a+b)^3\) – [3ab (a+b)].

Therefore, \(x^3\) + \((\frac{2}{x})^3\) =\( (x+\frac{2}{x})^3\) – [3 * x * \(\frac{2}{x}\) (x + \(\frac{2}{x}\))]. From the data given, x+\(\frac{2}{x}\) = 3. Substituting and simplifying, we get,

\(x^3\) + \((\frac{2}{x})^3\) = \((3)^3\) – [3*2(3)] = 27 – 18 = 9.

The correct answer option is C.

Hope that helps!

solve for \(x + \frac{2}{x} = 3\)
we get a quadratic and value of x= 2 &1
so \(x^3 + \frac{8}{x^3} =\)
value = 9
IMO C