Question

algebra: concepts and connections - plc introduction to polynomials which algebraic expression is a polynomial? $4x^2 - 3x + \frac{2}{x}$ $-2x^4 + \frac{3}{2x}$ $-6x^3 + x^2 - \sqrt{5}$ $8x^2 + \sqrt{x}$

Explanation:

Step1: Recall polynomial definition

A polynomial is an algebraic expression where variables have non-negative integer exponents, no variables in denominators, and no variables under radicals.

Step2: Analyze first expression

$4x^2 - 3x + \frac{2}{x}$ can be rewritten as $4x^2 - 3x + 2x^{-1}$. The term $2x^{-1}$ has a negative exponent, so this is not a polynomial.

Step3: Analyze second expression

$-2x^4 + \frac{3}{2x}$ can be rewritten as $-2x^4 + \frac{3}{2}x^{-1}$. The term $\frac{3}{2}x^{-1}$ has a negative exponent, so this is not a polynomial.

Step4: Analyze third expression

$-6x^3 + x^2 - \sqrt{5}$ has variables with non-negative integer exponents ($x^3$, $x^2$) and a constant term $-\sqrt{5}$. It meets all polynomial criteria.

Step5: Analyze fourth expression

$8x^2 + \sqrt{x}$ can be rewritten as $8x^2 + x^{\frac{1}{2}}$. The term $x^{\frac{1}{2}}$ has a non-integer exponent, so this is not a polynomial.

Answer:

$-6x^3 + x^2 - \sqrt{5}$