Get expert advice and community support for all your questions on IDNLearn.com. Discover reliable answers to your questions with our extensive database of expert knowledge.
Sagot :
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
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
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.