Question

The term independent of x in the expansion of [ (x + 1)/(x^{2/3} - x^{1/3} + 1) - (x - 1)/(x - x^{1/2}) ]¹⁰, x ≠ 1, is equal to ________.

Hint:

Look for algebraic identities like $a^3+b^3$ and $a^2-b^2$ within the terms to simplify binomial bases before expanding.

Answer 1

Correct Answer: 210

Step 1: Simplify the first term: $\frac{(x^{1/3})^3 + 1^3}{x^{2/3} - x^{1/3} + 1} = \frac{(x^{1/3}+1)(x^{2/3}-x^{1/3}+1)}{x^{2/3}-x^{1/3}+1} = x^{1/3} + 1$.
Step 2: Simplify the second term: $\frac{(\sqrt{x}-1)(\sqrt{x}+1)}{\sqrt{x}(\sqrt{x}-1)} = \frac{\sqrt{x}+1}{\sqrt{x}} = 1 + x^{-1/2}$.
Step 3: Expression becomes: $[(x^{1/3} + 1) - (1 + x^{-1/2})]^{10} = (x^{1/3} - x^{-1/2})^{10}$.
Step 4: General term $T_{r+1} = \binom{10}{r} (x^{1/3})^{10-r} (-x^{-1/2})^r = \binom{10}{r} (-1)^r x^{\frac{10-r}{3} - \frac{r}{2}}$.
Step 5: For independent term: $\frac{10-r}{3} - \frac{r}{2} = 0 \implies 20 - 2r - 3r = 0 \implies 5r = 20 \implies r = 4$.
Step 6: Value $= \binom{10}{4} (-1)^4 = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$.