If $x = 2 + 2^{\frac{2}{3}} + 2^{\frac{1}{3}}$, then the value of $x^3 - 6x^2 + 6x$ is

### Concept & Strategy When faced with evaluating a higher-degree polynomial given a complex surd value for $x$, direct substitution is highly inefficient. The strategy is to isolate the fractional power terms on one side, cube both sides to eliminate the fractions, and algebraically manipulate the result to match the target polynomial expression. $$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$$ ### Step-by-Step Solution * **Given:** $$x = 2 + 2^{\frac{2}{3}} + 2^{\frac{1}{3}}$$ Expression to find: $x^3 - 6x^2 + 6x$ * **Calculation / Deduction:** 1. Move the integer part to the left side to isolate the fractional powers: $$x - 2 = 2^{\frac{2}{3}} + 2^{\frac{1}{3}}$$ 2. Cube both sides to eliminate the fractional exponents: $$(x - 2)^3 = (2^{\frac{2}{3}} + 2^{\frac{1}{3}})^3$$ 3. Expand the left side using $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$: $$x^3 - 3(x^2)(2) + 3(x)(2^2) - 2^3 = x^3 - 6x^2 + 12x - 8$$ 4. Expand the right side using $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$: $$(2^{\frac{2}{3}})^3 + (2^{\frac{1}{3}})^3 + 3(2^{\frac{2}{3}})(2^{\frac{1}{3}})(2^{\frac{2}{3}} + 2^{\frac{1}{3}})$$ $$2^2 + 2^1 + 3(2^{\frac{2}{3} + \frac{1}{3}})(x - 2)$$ *(Note: We substituted $(x-2)$ back for the sum term)* $$4 + 2 + 3(2^1)(x - 2) = 6 + 6(x - 2) = 6 + 6x - 12 = 6x - 6$$ 5. Equate the expanded left and right sides: $$x^3 - 6x^2 + 12x - 8 = 6x - 6$$ 6. Rearrange the terms to match the required expression $x^3 - 6x^2 + 6x$: $$x^3 - 6x^2 + 12x - 6x = 8 - 6$$ $$x^3 - 6x^2 + 6x = 2$$ ### Exam Strategy & Shortcut **Pattern Recognition:** Questions of the format $x = a + a^{2/3} + a^{1/3}$ frequently ask for the value of $x^3 - 3ax^2 + 3a^2x$. Notice how the coefficients $-6$ and $+6$ in the target expression $x^3 - 6x^2 + 6x$ don't perfectly align with the standard binomial expansion (which yields $12x$). By carefully executing the "isolate and cube" method, you systematically force the target polynomial to appear on the left, leaving only a constant integer on the right. While there isn't a 5-second shortcut, recognizing this exact structural pattern prevents you from wasting time on brute-force substitution. ### Common Pitfall A major pitfall is expanding the right side $(2^{2/3} + 2^{1/3})^3$ and leaving the $3ab(a+b)$ term entirely in its fractional form. Failing to recognize that the $(a+b)$ component can be neatly replaced with $(x-2)$ leads to an algebraic dead end where the terms never simplify into integers. ### Final Answer **Therefore, the correct answer is $2$.**