Question
Which of these are polynomials
x^2 + 1
x^1/2 + 1
x/3
3/x
x^2 + 1
x^1/2 + 1
x/3
3/x
Answers
GPT-4o mini
To determine which of the given expressions are polynomials, we need to know the definition of a polynomial. A polynomial is an expression that consists of variables raised to non-negative integer powers, along with coefficients that can be real numbers.
Let's analyze each expression:
1. **\(x^2 + 1\)**: This is a polynomial because it is of the form \(ax^n\) where \(a\) is a coefficient (1 in this case), and \(n\) is a non-negative integer (2 here).
2. **\(x^{1/2} + 1\)**: This is not a polynomial because the exponent \(1/2\) is not a non-negative integer. This represents a radical (specifically the square root) rather than a polynomial term.
3. **\(x/3\)**: This can be rewritten as \(\frac{1}{3}x^1\). It is a polynomial because it is of the form \(ax^n\) with \(a = \frac{1}{3}\) and \(n = 1\), both of which meet the criteria.
4. **\(3/x\)**: This can be rewritten as \(3x^{-1}\). It is not a polynomial because the exponent \(-1\) is negative, which violates the criteria for a polynomial.
In summary, the polynomials from the list are:
- **\(x^2 + 1\)**
- **\(x/3\)**
The expressions \(x^{1/2} + 1\) and \(3/x\) are not polynomials.
Let's analyze each expression:
1. **\(x^2 + 1\)**: This is a polynomial because it is of the form \(ax^n\) where \(a\) is a coefficient (1 in this case), and \(n\) is a non-negative integer (2 here).
2. **\(x^{1/2} + 1\)**: This is not a polynomial because the exponent \(1/2\) is not a non-negative integer. This represents a radical (specifically the square root) rather than a polynomial term.
3. **\(x/3\)**: This can be rewritten as \(\frac{1}{3}x^1\). It is a polynomial because it is of the form \(ax^n\) with \(a = \frac{1}{3}\) and \(n = 1\), both of which meet the criteria.
4. **\(3/x\)**: This can be rewritten as \(3x^{-1}\). It is not a polynomial because the exponent \(-1\) is negative, which violates the criteria for a polynomial.
In summary, the polynomials from the list are:
- **\(x^2 + 1\)**
- **\(x/3\)**
The expressions \(x^{1/2} + 1\) and \(3/x\) are not polynomials.