The expansion $[x+(x^{3}-1)^{\frac{1}{2}}]^{5}+[x-(x^{3}-1)^{\frac{1}{2}}]^{5}$ is a polynomial of degree:

Question

Accepted Answer

To determine the degree of the polynomial given by the expression $[x+(x^{3}-1)^{\frac{1}{2}}]^{5}+[x-(x^{3}-1)^{\frac{1}{2}}]^{5}$, we need to analyze the structure of the expression.

### Explanation:

1. **Expression Analysis**:
   - The expression consists of two parts: $[x+(x^{3}-1)^{\frac{1}{2}}]^{5}$ and $[x-(x^{3}-1)^{\frac{1}{2}}]^{5}$.
   - Each part is raised to the power of 5, indicating that the maximum degree of any term in the expansion of each part could be 5.

2. **Symmetric Function**:
   - Notice that the expression is symmetric in the form of $a^5 + b^5$, where $a = x + (x^3 - 1)^{1/2}$ and $b = x - (x^3 - 1)^{1/2}$.
   - The symmetry suggests that certain terms will cancel out when the two expansions are added together.

3. **Degree Calculation**:
   - Consider the binomial expansion of each part:
     - The highest degree term in $[x+(x^{3}-1)^{\frac{1}{2}}]^{5}$ is $x^5$.
     - Similarly, the highest degree term in $[x-(x^{3}-1)^{\frac{1}{2}}]^{5}$ is also $x^5$.
   - However, due to the symmetry and the nature of the expression, the terms involving odd powers of $(x^3 - 1)^{1/2}$ will cancel out when the two expansions are added together.

4. **Resulting Polynomial Degree**:
   - The terms that survive after cancellation are those with even powers of $(x^3 - 1)^{1/2}$.
   - The highest degree term that does not cancel out is the one involving $x^3$ from $(x^3 - 1)^{1/2}$, which contributes to the degree of the polynomial.
   - Specifically, the term involving $(x^3 - 1)^{1/2}$ raised to the power of 2 will contribute a degree of 3, and when combined with the $x^2$ term from the binomial expansion, results in a term of degree 7.

5. **Conclusion**:
   - Therefore, the degree of the polynomial is 7, which corresponds to Option 3.

### Why Other Options are Incorrect:

- **Option 1 (Degree 5)**:
  - This option assumes that the highest degree term is simply $x^5$, ignoring the contribution from $(x^3 - 1)^{1/2}$.

- **Option 2 (Degree 6)**:
  - This option might consider partial contributions from $(x^3 - 1)^{1/2}$ but does not account for the full cancellation and resulting highest degree term.

- **Option 4 (Degree 8)**:
  - This option overestimates the degree by assuming that the highest degree terms from both expansions do not cancel out, which is not the case due to the symmetry and cancellation of odd powers.

In conclusion, the correct degree of the polynomial is 7, making Option 3 the correct choice.