Let's evaluate each algebraic expression to determine if it is a polynomial or not. A polynomial in one variable [tex]\( x \)[/tex] is an expression consisting of terms of the form [tex]\( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)[/tex], where [tex]\( n \)[/tex] is a non-negative integer and the coefficients [tex]\( a_i \)[/tex] are real numbers. The same applies for polynomials in more than one variable.
1. Expression: [tex]\( 2x^3 - \frac{1}{x} \)[/tex]
- This expression contains a term [tex]\( \frac{1}{x} \)[/tex] which is equivalent to [tex]\( x^{-1} \)[/tex].
- Since [tex]\( x^{-1} \)[/tex] involves a negative exponent, this term does not meet the criteria for polynomials.
- Conclusion: Not a polynomial.
2. Expression: [tex]\( x^3 y - 3x^2 + 6x \)[/tex]
- This expression consists of terms [tex]\( x^3 y \)[/tex], [tex]\( -3x^2 \)[/tex], and [tex]\( 6x \)[/tex].
- Even though the expression includes multiple variables, each variable is raised to a non-negative integer exponent.
- Conclusion: Polynomial.
3. Expression: [tex]\( y^2 + 5y - \sqrt{3} \)[/tex]
- This expression includes terms [tex]\( y^2 \)[/tex], [tex]\( 5y \)[/tex], and the constant term [tex]\( -\sqrt{3} \)[/tex].
- All exponents are non-negative integers and the constant term is a real number (irrational numbers are allowed).
- Conclusion: Polynomial.
4. Expression: [tex]\( 2 - \sqrt{4x} \)[/tex]
- This expression includes the term [tex]\( \sqrt{4x} \)[/tex], which can be rewritten as [tex]\( 2\sqrt{x} \)[/tex] or [tex]\( 2x^{1/2} \)[/tex].
- Since [tex]\( x^{1/2} \)[/tex] involves a fractional exponent, it does not meet the criteria for polynomials.
- Conclusion: Not a polynomial.
5. Expression: [tex]\( -x + \sqrt{6} \)[/tex]
- This expression includes terms [tex]\( -x \)[/tex] and the constant [tex]\( \sqrt{6} \)[/tex].
- Both terms meet the criteria for polynomials, as the exponent of [tex]\( x \)[/tex] is non-negative (specifically [tex]\( 1 \)[/tex]).
- Conclusion: Polynomial.
6. Expression: [tex]\( -\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4} \)[/tex]
- This expression consists of terms [tex]\( -\frac{1}{3} x^3 \)[/tex], [tex]\( -\frac{1}{2} x^2 \)[/tex], and the constant term [tex]\( \frac{1}{4} \)[/tex].
- All exponents are non-negative integers and the coefficients are real numbers.
- Conclusion: Polynomial.
Summarizing our conclusions:
- [tex]\( 2x^3 - \frac{1}{x} \)[/tex] → Not a polynomial
- [tex]\( x^3 y - 3x^2 + 6x \)[/tex] → Polynomial
- [tex]\( y^2 + 5y - \sqrt{3} \)[/tex] → Polynomial
- [tex]\( 2 - \sqrt{4x} \)[/tex] → Not a polynomial
- [tex]\( -x + \sqrt{6} \)[/tex] → Polynomial
- [tex]\( -\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4} \)[/tex] → Polynomial
