To determine which algebraic expression is a polynomial, we need to recall the definition of a polynomial. A polynomial in one variable [tex]\(x\)[/tex] is an expression that can be written in the form:
[tex]\[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \][/tex]
where the coefficients [tex]\(a_n, a_{n-1}, \ldots, a_1, a_0\)[/tex] are real numbers, and [tex]\(n\)[/tex] is a non-negative integer. It must only involve non-negative integer powers of [tex]\(x\)[/tex], and cannot include division by [tex]\(x\)[/tex] or roots.
Let's examine each given expression:
1. [tex]\( 4 x^2 - 3 x + \frac{2}{x} \)[/tex]
- This expression includes the term [tex]\(\frac{2}{x}\)[/tex], which involves division by the variable [tex]\(x\)[/tex]. Hence, it is not a polynomial.
2. [tex]\( -6 x^3 + x^2 - \sqrt{5} \)[/tex]
- This expression involves [tex]\(\sqrt{5}\)[/tex], but note that [tex]\(\sqrt{5}\)[/tex] is just a constant and doesn't affect the form as a polynomial. Thus, this expression is not problematic because it only combines terms with non-negative integer powers of [tex]\(x\)[/tex]. So it is indeed a polynomial.
3. [tex]\( 8 x^2 + \sqrt{x} \)[/tex]
- The term [tex]\(\sqrt{x}\)[/tex] can be rewritten as [tex]\(x^{1/2}\)[/tex], which is not a non-negative integer power of [tex]\(x\)[/tex]. Hence, this expression is not a polynomial.
4. [tex]\( -2 x^4 + \frac{3}{2 x} \)[/tex]
- This expression contains the term [tex]\(\frac{3}{2 x}\)[/tex], which involves division by the variable [tex]\(x\)[/tex]. Thus, it is not a polynomial.
Upon inspecting each of the expressions based on the criteria for a polynomial, we have:
The algebraic expression that is a polynomial is:
\[ -6 x^3 + x^2 - \sqrt{5} \ ]
Thus, the polynomial among the given expressions is the second one.
