To solve this problem, we need to determine which of the given mathematical expressions are polynomials. A polynomial is defined as an expression that consists of variables raised to non-negative integer powers, with coefficients that can be real or complex numbers. Here are the expressions we're evaluating:
1. [tex]\(\frac{1}{5} x^3 - \frac{7}{2} x^2 - 2\)[/tex]
2. [tex]\(5x^2 - 2x + 7\)[/tex]
3. [tex]\(\frac{y^3}{3} - \sqrt{y} + 2\)[/tex]
4. [tex]\(y^5 - \sqrt{5} y^2 + 7\)[/tex]
We will evaluate these one by one to determine if they meet the criteria for being polynomials.
### Expression 1: [tex]\(\frac{1}{5} x^3 - \frac{7}{2} x^2 - 2\)[/tex]
- Each term in this expression is either a constant or a variable raised to a non-negative integer power.
- [tex]\(\frac{1}{5} x^3\)[/tex] has [tex]\(x\)[/tex] raised to the power of 3 (an integer power).
- [tex]\(- \frac{7}{2} x^2\)[/tex] has [tex]\(x\)[/tex] raised to the power of 2 (an integer power).
- [tex]\(- 2\)[/tex] is a constant.
Since this expression only includes variables raised to non-negative integer powers, it is a polynomial.
### Expression 2: [tex]\(5x^2 - 2x + 7\)[/tex]
- Each term in this expression is either a constant or a variable raised to a non-negative integer power.
- [tex]\(5x^2\)[/tex] has [tex]\(x\)[/tex] raised to the power of 2 (an integer power).
- [tex]\(-2x\)[/tex] has [tex]\(x\)[/tex] raised to the power of 1 (an integer power).
- [tex]\(7\)[/tex] is a constant.
Since this expression only includes variables raised to non-negative integer powers, it is a polynomial.
### Expression 3: [tex]\(\frac{y^3}{3} - \sqrt{y} + 2\)[/tex]
- The term [tex]\(\frac{y^3}{3}\)[/tex] has [tex]\(y\)[/tex] raised to the power of 3 (an integer power).
- The term [tex]\(\sqrt{y}\)[/tex] can be rewritten as [tex]\(y^{1/2}\)[/tex], which is a non-integer power.
- The term [tex]\(2\)[/tex] is a constant.
One of the terms involves [tex]\(y\)[/tex] raised to a non-integer power ([tex]\(y^{1/2}\)[/tex]), so this expression is not a polynomial.
### Expression 4: [tex]\(y^5 - \sqrt{5} y^2 + 7\)[/tex]
- The term [tex]\(y^5\)[/tex] has [tex]\(y\)[/tex] raised to the power of 5 (an integer power).
- The term [tex]\( \sqrt{5} y^2\)[/tex] has [tex]\(y\)[/tex] raised to the power of 2 (an integer power), and [tex]\(\sqrt{5}\)[/tex] is a constant coefficient.
- The term [tex]\(7\)[/tex] is a constant.
Since this expression only consists of variables raised to non-negative integer powers, it is a polynomial.
### Conclusion
Thus, the expressions that are polynomials are:
- [tex]\(\frac{1}{5} x^3 - \frac{7}{2} x^2 - 2\)[/tex]
- [tex]\(5x^2 - 2x + 7\)[/tex]
- [tex]\(y^5 - \sqrt{5} y^2 + 7\)[/tex]
So, the correct answers are the first, second, and fourth expressions.
