Answer :
To determine which expression is a polynomial, let's examine each option individually:
1. Polynomial Definition:
- A polynomial is an expression that consists of variables and coefficients, involving only non-negative integer exponents of variables. It does not include operations like division by a variable, logarithms, or negative exponents.
Let's evaluate each option:
A. Expression: [tex]\(x^3 + 6x - \log x\)[/tex]
- This expression includes the term [tex]\(\log x\)[/tex]. Since logarithms are not allowed in polynomials, this is not a polynomial expression.
B. Expression: [tex]\(\frac{4x^4 - 16x^3 - x^2}{\pi x^3}\)[/tex]
- Simplifying this expression involves dividing each term in the numerator by [tex]\(x^3\)[/tex]. This results in terms with negative exponents, which are not allowed in polynomials. Thus, this is not a polynomial expression.
C. Expression: [tex]\(4x^2 - 16x^3 - x^{-1}\)[/tex]
- This expression includes the term [tex]\(x^{-1}\)[/tex], which is a negative exponent. Negative exponents are not allowed in polynomials, so this is not a polynomial expression.
D. None of these
- Since none of the options A, B, or C meet the criteria for being a polynomial, the correct choice is D: None of these.
Therefore, the expression that represents a polynomial is: None of these (D).
1. Polynomial Definition:
- A polynomial is an expression that consists of variables and coefficients, involving only non-negative integer exponents of variables. It does not include operations like division by a variable, logarithms, or negative exponents.
Let's evaluate each option:
A. Expression: [tex]\(x^3 + 6x - \log x\)[/tex]
- This expression includes the term [tex]\(\log x\)[/tex]. Since logarithms are not allowed in polynomials, this is not a polynomial expression.
B. Expression: [tex]\(\frac{4x^4 - 16x^3 - x^2}{\pi x^3}\)[/tex]
- Simplifying this expression involves dividing each term in the numerator by [tex]\(x^3\)[/tex]. This results in terms with negative exponents, which are not allowed in polynomials. Thus, this is not a polynomial expression.
C. Expression: [tex]\(4x^2 - 16x^3 - x^{-1}\)[/tex]
- This expression includes the term [tex]\(x^{-1}\)[/tex], which is a negative exponent. Negative exponents are not allowed in polynomials, so this is not a polynomial expression.
D. None of these
- Since none of the options A, B, or C meet the criteria for being a polynomial, the correct choice is D: None of these.
Therefore, the expression that represents a polynomial is: None of these (D).
