To determine which of the given expressions are polynomials, we need to check if each term in the expression satisfies the definition of a polynomial. A polynomial is an expression consisting of terms that are non-negative integer powers of the variable [tex]\(x\)[/tex], with coefficients that are real numbers.
Let's analyze each option:
### Option A: [tex]\(x^{-2} + 15x - 3\)[/tex]
- The first term is [tex]\(x^{-2}\)[/tex], which is [tex]\(x\)[/tex] raised to a negative exponent.
- Since polynomials do not include terms with negative exponents, this expression is not a polynomial.
### Option B: [tex]\(\frac{3}{5}x^4 - 18x^2 + 5 - \frac{10}{x^2}\)[/tex]
- The first term, [tex]\(\frac{3}{5}x^4\)[/tex], is acceptable as it is a positive integer power of [tex]\(x\)[/tex] multiplied by a coefficient.
- The second term, [tex]\(-18x^2\)[/tex], is also acceptable.
- The third term, [tex]\(5\)[/tex], is a constant term which is acceptable.
- The fourth term, [tex]\(\frac{10}{x^2}\)[/tex], can be rewritten as [tex]\(10x^{-2}\)[/tex] which includes a negative exponent.
- Since this negative exponent exists, this expression is not a polynomial.
### Option C: [tex]\(4x^4 - 10\)[/tex]
- The first term, [tex]\(4x^4\)[/tex], is a positive integer power of [tex]\(x\)[/tex] multiplied by a coefficient.
- The second term, [tex]\(-10\)[/tex], is a constant term.
- All terms fit the definition of a polynomial.
- Therefore, this expression is a polynomial.
### Option D: [tex]\(5.3x^2 + 3x - 2\)[/tex]
- The first term, [tex]\(5.3x^2\)[/tex], is a positive integer power of [tex]\(x\)[/tex] multiplied by a coefficient.
- The second term, [tex]\(3x\)[/tex], is a positive integer power of [tex]\(x\)[/tex] multiplied by a coefficient.
- The third term, [tex]\(-2\)[/tex], is a constant term.
- All terms fit the definition of a polynomial.
- Therefore, this expression is a polynomial.
### Option E: [tex]\(-x^3 + 5x^2 + 7\sqrt{x} - 1\)[/tex]
- The first term, [tex]\(-x^3\)[/tex], is a positive integer power of [tex]\(x\)[/tex] multiplied by a coefficient.
- The second term, [tex]\(5x^2\)[/tex], is also a positive integer power of [tex]\(x\)[/tex] multiplied by a coefficient.
- The third term, [tex]\(7\sqrt{x}\)[/tex], can be rewritten as [tex]\(7x^{1/2}\)[/tex] which includes a non-integer exponent.
- The fourth term, [tex]\(-1\)[/tex], is a constant term.
- The presence of the term [tex]\(7x^{1/2}\)[/tex] with a non-integer exponent means this expression is not a polynomial.
### Conclusion
The expressions that are polynomials are:
- Option C: [tex]\(4x^4 - 10\)[/tex]
- Option D: [tex]\(5.3x^2 + 3x - 2\)[/tex]
So, the correct answers are C and D.
