Answer :
To determine which of the given algebraic expressions are polynomials, we need to understand the definition of a polynomial. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
Let's analyze each given expression step by step:
### Expression 1: [tex]\( 2x^3 - \frac{1}{x} \)[/tex]
- Terms: [tex]\( 2x^3 \)[/tex] and [tex]\( -\frac{1}{x} \)[/tex]
- Analysis: The term [tex]\( 2x^3 \)[/tex] is a polynomial term because it involves [tex]\( x \)[/tex] raised to a non-negative integer power (3). However, the term [tex]\( -\frac{1}{x} \)[/tex] is not a polynomial term because it can be written as [tex]\( -x^{-1} \)[/tex] and involves a negative exponent.
- Conclusion: This expression is not a polynomial.
### Expression 2: [tex]\( x^3 y - 3x^2 + 6x \)[/tex]
- Terms: [tex]\( x^3 y \)[/tex], [tex]\( -3x^2 \)[/tex], and [tex]\( 6x \)[/tex]
- Analysis: The term [tex]\( x^3 y \)[/tex] involves two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] raised to non-negative integer powers. The terms [tex]\( -3x^2 \)[/tex] and [tex]\( 6x \)[/tex] are typical polynomial terms.
- Conclusion: This expression is a polynomial in [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Expression 3: [tex]\( y^2 + 5 y - \sqrt{3} \)[/tex]
- Terms: [tex]\( y^2 \)[/tex], [tex]\( 5y \)[/tex], and [tex]\( -\sqrt{3} \)[/tex]
- Analysis: Each term involves [tex]\( y \)[/tex] raised to a non-negative integer power: [tex]\( y^2 \)[/tex] and [tex]\( 5y \)[/tex]. The constant term [tex]\( -\sqrt{3} \)[/tex], though an irrational number, does not affect the polynomial nature.
- Conclusion: This expression is a polynomial.
### Expression 4: [tex]\( 2 - \sqrt{4x} \)[/tex]
- Terms: [tex]\( 2 \)[/tex] and [tex]\( -\sqrt{4x} \)[/tex]
- Analysis: The term [tex]\( 2 \)[/tex] is a constant and can be considered a polynomial term. The term [tex]\( -\sqrt{4x} \)[/tex] can be rewritten as [tex]\( -2\sqrt{x} \)[/tex], which involves [tex]\( x \)[/tex] raised to the power of [tex]\( \frac{1}{2} \)[/tex], a non-integer.
- Conclusion: This expression is not a polynomial.
### Expression 5: [tex]\( -x + \sqrt{6} \)[/tex]
- Terms: [tex]\( -x \)[/tex] and [tex]\( \sqrt{6} \)[/tex]
- Analysis: The term [tex]\( -x \)[/tex] is a polynomial term with [tex]\( x \)[/tex] raised to the power of 1. The constant term [tex]\( \sqrt{6} \)[/tex] does not involve any variables.
- Conclusion: This expression is a polynomial.
### Expression 6: [tex]\( -\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4} \)[/tex]
- Terms: [tex]\( -\frac{1}{3} x^3 \)[/tex], [tex]\( -\frac{1}{2} x^2 \)[/tex], and [tex]\( \frac{1}{4} \)[/tex]
- Analysis: Each term involves [tex]\( x \)[/tex] raised to non-negative integer powers: [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex]. The constant term [tex]\( \frac{1}{4} \)[/tex] is a standard part of a polynomial.
- Conclusion: This expression is a polynomial.
### Summary
The expressions which are polynomials are:
- [tex]\( x^3 y - 3x^2 + 6x \)[/tex]
- [tex]\( y^2 + 5 y - \sqrt{3} \)[/tex]
- [tex]\( -x + \sqrt{6} \)[/tex]
- [tex]\( -\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4} \)[/tex]
The expressions which are not polynomials are:
- [tex]\( 2x^3 - \frac{1}{x} \)[/tex]
- [tex]\( 2 - \sqrt{4x} \)[/tex]
Let's analyze each given expression step by step:
### Expression 1: [tex]\( 2x^3 - \frac{1}{x} \)[/tex]
- Terms: [tex]\( 2x^3 \)[/tex] and [tex]\( -\frac{1}{x} \)[/tex]
- Analysis: The term [tex]\( 2x^3 \)[/tex] is a polynomial term because it involves [tex]\( x \)[/tex] raised to a non-negative integer power (3). However, the term [tex]\( -\frac{1}{x} \)[/tex] is not a polynomial term because it can be written as [tex]\( -x^{-1} \)[/tex] and involves a negative exponent.
- Conclusion: This expression is not a polynomial.
### Expression 2: [tex]\( x^3 y - 3x^2 + 6x \)[/tex]
- Terms: [tex]\( x^3 y \)[/tex], [tex]\( -3x^2 \)[/tex], and [tex]\( 6x \)[/tex]
- Analysis: The term [tex]\( x^3 y \)[/tex] involves two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] raised to non-negative integer powers. The terms [tex]\( -3x^2 \)[/tex] and [tex]\( 6x \)[/tex] are typical polynomial terms.
- Conclusion: This expression is a polynomial in [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Expression 3: [tex]\( y^2 + 5 y - \sqrt{3} \)[/tex]
- Terms: [tex]\( y^2 \)[/tex], [tex]\( 5y \)[/tex], and [tex]\( -\sqrt{3} \)[/tex]
- Analysis: Each term involves [tex]\( y \)[/tex] raised to a non-negative integer power: [tex]\( y^2 \)[/tex] and [tex]\( 5y \)[/tex]. The constant term [tex]\( -\sqrt{3} \)[/tex], though an irrational number, does not affect the polynomial nature.
- Conclusion: This expression is a polynomial.
### Expression 4: [tex]\( 2 - \sqrt{4x} \)[/tex]
- Terms: [tex]\( 2 \)[/tex] and [tex]\( -\sqrt{4x} \)[/tex]
- Analysis: The term [tex]\( 2 \)[/tex] is a constant and can be considered a polynomial term. The term [tex]\( -\sqrt{4x} \)[/tex] can be rewritten as [tex]\( -2\sqrt{x} \)[/tex], which involves [tex]\( x \)[/tex] raised to the power of [tex]\( \frac{1}{2} \)[/tex], a non-integer.
- Conclusion: This expression is not a polynomial.
### Expression 5: [tex]\( -x + \sqrt{6} \)[/tex]
- Terms: [tex]\( -x \)[/tex] and [tex]\( \sqrt{6} \)[/tex]
- Analysis: The term [tex]\( -x \)[/tex] is a polynomial term with [tex]\( x \)[/tex] raised to the power of 1. The constant term [tex]\( \sqrt{6} \)[/tex] does not involve any variables.
- Conclusion: This expression is a polynomial.
### Expression 6: [tex]\( -\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4} \)[/tex]
- Terms: [tex]\( -\frac{1}{3} x^3 \)[/tex], [tex]\( -\frac{1}{2} x^2 \)[/tex], and [tex]\( \frac{1}{4} \)[/tex]
- Analysis: Each term involves [tex]\( x \)[/tex] raised to non-negative integer powers: [tex]\( x^3 \)[/tex] and [tex]\( x^2 \)[/tex]. The constant term [tex]\( \frac{1}{4} \)[/tex] is a standard part of a polynomial.
- Conclusion: This expression is a polynomial.
### Summary
The expressions which are polynomials are:
- [tex]\( x^3 y - 3x^2 + 6x \)[/tex]
- [tex]\( y^2 + 5 y - \sqrt{3} \)[/tex]
- [tex]\( -x + \sqrt{6} \)[/tex]
- [tex]\( -\frac{1}{3} x^3 - \frac{1}{2} x^2 + \frac{1}{4} \)[/tex]
The expressions which are not polynomials are:
- [tex]\( 2x^3 - \frac{1}{x} \)[/tex]
- [tex]\( 2 - \sqrt{4x} \)[/tex]
