To determine if the expression \( \frac{1}{2}x^3 - \sqrt{3}x + 1 \) is a polynomial, we need to consider the definition of a polynomial. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients, where the coefficients are usually real numbers.
In this expression:
- \( \frac{1}{2}x^3 \) is a polynomial term since it has a non-negative integer exponent (3).
- \( -\sqrt{3}x \) is also a polynomial term because it has a non-negative integer exponent (1) and the coefficient \( -\sqrt{3} \) is a real number.
- \( 1 \) is a polynomial term as well, corresponding to \( x^0 \).
Since all the terms in the expression are polynomial terms with real coefficients, the entire expression \( \frac{1}{2}x^3 - \sqrt{3}x + 1 \) is indeed a polynomial.
Thus, the expression is a polynomial.