Determine if the function reprsents a polynomial, and for those that are state their degree. 13) -2a^5+10a^6 14) 4x^6 15) x^4cdot (x-1)^2cdot (x+2.5)^3 16) sqrt (x)+1 17) (x^2+2)/(3) 18) x^(1)/(3)+7x^2-2 19) (x+2)^3cdot (x-(3)/(5))^2 20) 3x^2+7x-(2)/(3x) 21) (x-(1)/(4))^4cdot (x+sqrt (7))^2 22) (x+1)/(x^2)

## Step 1: Analyzing the terms of each expression<br />### We examine each term of the given expressions to determine if they fit the definition of a polynomial. A polynomial has only non-negative integer exponents on its variables.<br /><br />## Step 2: Identifying Polynomials and their Degrees<br />### For expressions that are polynomials, we identify the highest power of the variable, which represents the degree of the polynomial.<br /><br />## Step 3: Results for each expression<br />### 13) $-2a^{5}+10a^{6}$: This is a polynomial. The highest power is 6, so the degree is 6.<br />### 14) $4x^{6}$: This is a polynomial. The highest power is 6, so the degree is 6.<br />### 15) $x^{4}\cdot (x-1)^{2}\cdot (x+2.5)^{3}$: This is a polynomial. Expanding the expression would result in a highest power of $4+2+3=9$, so the degree is 9.<br />### 16) $\sqrt {x}+1$: This is not a polynomial because $\sqrt{x} = x^{\frac{1}{2}}$ and $\frac{1}{2}$ is not an integer.<br />### 17) $\frac {x^{2}+2}{3}$: This is a polynomial. It can be rewritten as $\frac{1}{3}x^2 + \frac{2}{3}$. The highest power is 2, so the degree is 2.<br />### 18) $x^{\frac {1}{3}}+7x^{2}-2$: This is not a polynomial because $x^{\frac{1}{3}}$ has a non-integer exponent.<br />### 19) $(x+2)^{3}\cdot (x-\frac {3}{5})^{2}$: This is a polynomial. Expanding the expression would result in a highest power of $3+2=5$, so the degree is 5.<br />### 20) $3x^{2}+7x-\frac {2}{3x}$: This is not a polynomial because $\frac{2}{3x} = \frac{2}{3}x^{-1}$ and -1 is a negative exponent.<br />### 21) $(x-\frac {1}{4})^{4}\cdot (x+\sqrt {7})^{2}$: This is a polynomial. Expanding the expression would result in a highest power of $4+2=6$, so the degree is 6.<br />### 22) $\frac {x+1}{x^{2}}$: This is not a polynomial because $\frac{x+1}{x^2} = x^{-1} + x^{-2}$, which contains negative exponents.