Get the information you need from a community of experts on IDNLearn.com. Find reliable solutions to your questions quickly and easily with help from our experienced experts.
Sagot :
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.
