Identify which of the following expressions are polynomials. If so, write their degrees.
`8x^(2)-3x+6sqrt(x)+1`

Answer

To determine whether the expression \(8x^{2} - 3x + 6\sqrt{x} + 1\) is a polynomial and to find its degree, we can follow these steps:

### Step 1: Identify the components of the expression
The expression given is:
\[ 8x^{2} - 3x + 6\sqrt{x} + 1 \]

### Step 2: Check each term to see if they are polynomial terms
A polynomial is defined as an expression that consists of variables raised to non-negative integer powers. Let's analyze each term:
- \(8x^{2}\): This term is a polynomial term because the exponent (2) is a non-negative integer.
- \(-3x\): This term is also a polynomial term because the exponent (1) is a non-negative integer.
- \(6\sqrt{x}\): This term can be rewritten as \(6x^{1/2}\). Since the exponent (1/2) is not a non-negative integer, this term does not qualify as a polynomial term.
- \(+1\): This term is a polynomial term because it can be considered as \(1x^{0}\) (where the exponent is 0).

### Step 3: Determine if the entire expression is a polynomial
Since one of the terms, \(6\sqrt{x}\), is not a polynomial term, the entire expression \(8x^{2} - 3x + 6\sqrt{x} + 1\) is **not** a polynomial.

### Step 4: Conclusion
The expression \(8x^{2} - 3x + 6\sqrt{x} + 1\) is not a polynomial.