Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine which of the given algebraic expressions is a polynomial, we need to understand the definition of a polynomial.
A polynomial in a single variable [tex]\( x \)[/tex] is an algebraic expression that consists of terms in the form [tex]\( a_n x^n \)[/tex], where [tex]\( a_n \)[/tex] is a coefficient and [tex]\( n \)[/tex] is a non-negative integer. Polynomials can be added, subtracted, and multiplied together, and they can also be scaled by multiplying or dividing by constants.
Given this, let's analyze each expression one by one:
1. [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex]
- [tex]\( 4x^2 \)[/tex] is a polynomial term.
- [tex]\( -3x \)[/tex] is a polynomial term.
- [tex]\( \frac{2}{x} = 2x^{-1} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\(-1\)[/tex], which is not a non-negative integer.
Hence, [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex] is not a polynomial.
2. [tex]\( -6x^3 + x^2 - \sqrt{5} \)[/tex]
- [tex]\( -6x^3 \)[/tex] is a polynomial term.
- [tex]\( x^2 \)[/tex] is a polynomial term.
- [tex]\( -\sqrt{5} \)[/tex] is a constant term (coefficients can be constants or roots), which means it is part of a polynomial.
Hence, [tex]\( -6x^3 + x^2 - \sqrt{5} \)[/tex] is a polynomial.
3. [tex]\( 8x^2 + \sqrt{x} \)[/tex]
- [tex]\( 8x^2 \)[/tex] is a polynomial term.
- [tex]\( \sqrt{x} = x^{1/2} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\( \frac{1}{2} \)[/tex], which is not an integer.
Hence, [tex]\( 8x^2 + \sqrt{x} \)[/tex] is not a polynomial.
4. [tex]\( -2x^4 + \frac{3}{2x} \)[/tex]
- [tex]\( -2x^4 \)[/tex] is a polynomial term.
- [tex]\( \frac{3}{2x} = \frac{3}{2} x^{-1} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\(-1\)[/tex], which is not a non-negative integer.
Hence, [tex]\( -2x^4 + \frac{3}{2x} \)[/tex] is not a polynomial.
In conclusion, out of the given algebraic expressions, the only one that is a polynomial is:
[tex]\[ -6x^3 + x^2 - \sqrt{5} \][/tex]
A polynomial in a single variable [tex]\( x \)[/tex] is an algebraic expression that consists of terms in the form [tex]\( a_n x^n \)[/tex], where [tex]\( a_n \)[/tex] is a coefficient and [tex]\( n \)[/tex] is a non-negative integer. Polynomials can be added, subtracted, and multiplied together, and they can also be scaled by multiplying or dividing by constants.
Given this, let's analyze each expression one by one:
1. [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex]
- [tex]\( 4x^2 \)[/tex] is a polynomial term.
- [tex]\( -3x \)[/tex] is a polynomial term.
- [tex]\( \frac{2}{x} = 2x^{-1} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\(-1\)[/tex], which is not a non-negative integer.
Hence, [tex]\( 4x^2 - 3x + \frac{2}{x} \)[/tex] is not a polynomial.
2. [tex]\( -6x^3 + x^2 - \sqrt{5} \)[/tex]
- [tex]\( -6x^3 \)[/tex] is a polynomial term.
- [tex]\( x^2 \)[/tex] is a polynomial term.
- [tex]\( -\sqrt{5} \)[/tex] is a constant term (coefficients can be constants or roots), which means it is part of a polynomial.
Hence, [tex]\( -6x^3 + x^2 - \sqrt{5} \)[/tex] is a polynomial.
3. [tex]\( 8x^2 + \sqrt{x} \)[/tex]
- [tex]\( 8x^2 \)[/tex] is a polynomial term.
- [tex]\( \sqrt{x} = x^{1/2} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\( \frac{1}{2} \)[/tex], which is not an integer.
Hence, [tex]\( 8x^2 + \sqrt{x} \)[/tex] is not a polynomial.
4. [tex]\( -2x^4 + \frac{3}{2x} \)[/tex]
- [tex]\( -2x^4 \)[/tex] is a polynomial term.
- [tex]\( \frac{3}{2x} = \frac{3}{2} x^{-1} \)[/tex] is not a polynomial term because the exponent of [tex]\( x \)[/tex] is [tex]\(-1\)[/tex], which is not a non-negative integer.
Hence, [tex]\( -2x^4 + \frac{3}{2x} \)[/tex] is not a polynomial.
In conclusion, out of the given algebraic expressions, the only one that is a polynomial is:
[tex]\[ -6x^3 + x^2 - \sqrt{5} \][/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
