Determine if the expression \( -6 r^{2} \sqrt{6}+\frac{1}{r^{3}}-8 r^{4} \) is a polynomial or not. If it is a polynomial, state the type and degree of the polynomial. Answer Attempt 1 out of 2 The given expression \( \sim \) a polynomial.

To determine whether the expression \[ -6 r^{2} \sqrt{6} + \frac{1}{r^{3}} - 8 r^{4} \] is a polynomial, we need to examine each term in the expression. **Definition of a Polynomial:** A polynomial in variable \( r \) is an expression consisting of terms with non-negative integer exponents of \( r \). Each term is of the form \( a_n r^n \), where \( a_n \) is a constant coefficient and \( n \) is a non-negative integer. **Analyzing Each Term:** 1. **\(-6 r^{2} \sqrt{6}\):** - This term can be rewritten as \(-6 \sqrt{6} \cdot r^{2}\). - The exponent of \( r \) is \( 2 \), which is a non-negative integer. 2. **\(\frac{1}{r^{3}}\):** - This term can be rewritten as \( r^{-3} \). - The exponent of \( r \) is \( -3 \), which is a negative integer. 3. **\(-8 r^{4}\):** - The exponent of \( r \) is \( 4 \), which is a non-negative integer. **Conclusion:** - The presence of the term \( \frac{1}{r^{3}} \) (or \( r^{-3} \)) introduces a negative exponent. - Since polynomials require all exponents to be non-negative integers, this term violates the definition of a polynomial. **Therefore, the given expression is **_not_** a polynomial**. **Final Answer:** No. Because it includes a term with a negative exponent, the expression is not a polynomial.