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To determine if an algebraic expression is a polynomial, we check each term to see if it meets the criteria for a polynomial. Specifically, all variables must have non-negative integer exponents, and the expression must not contain any roots of variables, nor variables in the denominator.
Let's analyze each expression in detail:
1. Expression: [tex]\( 2x^3 - \frac{1}{x} \)[/tex]
- The term [tex]\(2x^3\)[/tex] is a valid polynomial term because the exponent of [tex]\(x\)[/tex] is 3, a non-negative integer.
- The term [tex]\(\frac{1}{x}\)[/tex] can be written as [tex]\(x^{-1}\)[/tex], here, the exponent is -1, which is a negative integer.
Since the term [tex]\( \frac{1}{x} \)[/tex] invalidates the criteria for a polynomial, the expression [tex]\( 2x^3 - \frac{1}{x} \)[/tex] is not a polynomial.
2. Expression: [tex]\( x^3 y - 3x^2 + 6x \)[/tex]
- The term [tex]\( x^3 y \)[/tex] involves [tex]\( y \)[/tex] with an implied exponent of 1, but neither exponent is negative or a fraction.
- The term [tex]\( -3x^2 \)[/tex] has exponent 2, a non-negative integer.
- The term [tex]\( 6x \)[/tex] has exponent 1, a non-negative integer.
However, because the expression involves a product of variables (i.e., [tex]\( x^3 y \)[/tex]), it does not meet the strict polynomial definition, so [tex]\( x^3 y - 3x^2 + 6x \)[/tex] is not a polynomial.
3. Expression: [tex]\( y^2 + 5y - \sqrt{3} \)[/tex]
- The term [tex]\( y^2 \)[/tex] has exponent 2, a non-negative integer.
- The term [tex]\( 5y \)[/tex] has exponent 1, a non-negative integer.
- The term [tex]\( -\sqrt{3} \)[/tex] is a constant term and doesn't affect whether the expression is a polynomial.
Since all variable terms have non-negative integer exponents, [tex]\( y^2 + 5y - \sqrt{3} \)[/tex] is a polynomial.
4. Expression: [tex]\( 2 - \sqrt{4x} \)[/tex]
- The term [tex]\( 2 \)[/tex] is a constant and doesn't affect whether the expression is a polynomial.
- The term [tex]\(\sqrt{4x}\)[/tex] can be written as [tex]\((4x)^{1/2}\)[/tex]. Here, the exponent of [tex]\(x\)[/tex] is 1/2, which is a fraction.
Since the term [tex]\(\sqrt{4x}\)[/tex] invalidates the criteria for a polynomial, the expression [tex]\( 2 - \sqrt{4x} \)[/tex] is not a polynomial.
5. Expression: [tex]\( -x + \sqrt{6} \)[/tex]
- The term [tex]\( -x \)[/tex] has exponent 1, a non-negative integer.
- The term [tex]\( \sqrt{6} \)[/tex] is a constant term and doesn’t affect whether the expression is a polynomial.
Since all variable terms have non-negative integer exponents, [tex]\( -x + \sqrt{6} \)[/tex] is a polynomial.
To summarize, none of the given expressions are polynomials.
Let's analyze each expression in detail:
1. Expression: [tex]\( 2x^3 - \frac{1}{x} \)[/tex]
- The term [tex]\(2x^3\)[/tex] is a valid polynomial term because the exponent of [tex]\(x\)[/tex] is 3, a non-negative integer.
- The term [tex]\(\frac{1}{x}\)[/tex] can be written as [tex]\(x^{-1}\)[/tex], here, the exponent is -1, which is a negative integer.
Since the term [tex]\( \frac{1}{x} \)[/tex] invalidates the criteria for a polynomial, the expression [tex]\( 2x^3 - \frac{1}{x} \)[/tex] is not a polynomial.
2. Expression: [tex]\( x^3 y - 3x^2 + 6x \)[/tex]
- The term [tex]\( x^3 y \)[/tex] involves [tex]\( y \)[/tex] with an implied exponent of 1, but neither exponent is negative or a fraction.
- The term [tex]\( -3x^2 \)[/tex] has exponent 2, a non-negative integer.
- The term [tex]\( 6x \)[/tex] has exponent 1, a non-negative integer.
However, because the expression involves a product of variables (i.e., [tex]\( x^3 y \)[/tex]), it does not meet the strict polynomial definition, so [tex]\( x^3 y - 3x^2 + 6x \)[/tex] is not a polynomial.
3. Expression: [tex]\( y^2 + 5y - \sqrt{3} \)[/tex]
- The term [tex]\( y^2 \)[/tex] has exponent 2, a non-negative integer.
- The term [tex]\( 5y \)[/tex] has exponent 1, a non-negative integer.
- The term [tex]\( -\sqrt{3} \)[/tex] is a constant term and doesn't affect whether the expression is a polynomial.
Since all variable terms have non-negative integer exponents, [tex]\( y^2 + 5y - \sqrt{3} \)[/tex] is a polynomial.
4. Expression: [tex]\( 2 - \sqrt{4x} \)[/tex]
- The term [tex]\( 2 \)[/tex] is a constant and doesn't affect whether the expression is a polynomial.
- The term [tex]\(\sqrt{4x}\)[/tex] can be written as [tex]\((4x)^{1/2}\)[/tex]. Here, the exponent of [tex]\(x\)[/tex] is 1/2, which is a fraction.
Since the term [tex]\(\sqrt{4x}\)[/tex] invalidates the criteria for a polynomial, the expression [tex]\( 2 - \sqrt{4x} \)[/tex] is not a polynomial.
5. Expression: [tex]\( -x + \sqrt{6} \)[/tex]
- The term [tex]\( -x \)[/tex] has exponent 1, a non-negative integer.
- The term [tex]\( \sqrt{6} \)[/tex] is a constant term and doesn’t affect whether the expression is a polynomial.
Since all variable terms have non-negative integer exponents, [tex]\( -x + \sqrt{6} \)[/tex] is a polynomial.
To summarize, none of the given expressions are polynomials.
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