which algebraic expression is a polynomial with a degree of 4?\n$5x^{4}+sqrt{4x}$\n$x^{5}-6x^{4}+14x^{3}+x^{2…

Explanation:

Step1: Recall polynomial definition

A polynomial in one - variable $x$ is an expression of the form $a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0$, where $n$ is a non - negative integer and $a_i$ are constants. The degree of a polynomial is the highest power of the variable with a non - zero coefficient. Also, polynomials do not have variables in the denominator or under a radical.

Step2: Analyze option 1

For the expression $5x^4+\sqrt{4x}=5x^4 + 2\sqrt{x}$, since it has a square - root of the variable $x$, it is not a polynomial.

Step3: Analyze option 2

For the expression $x^5-6x^4 + 14x^3+x^2$, the highest power of $x$ is $5$, so its degree is $5$.

Step4: Analyze option 3

For the expression $9x^4-x^3-\frac{x}{5}=9x^4 - x^3-\frac{1}{5}x$, the highest power of $x$ is $4$, and it is in the form of a polynomial (no variables in the denominator or under a radical).

Step5: Analyze option 4

For the expression $2x^4-6x^4+\frac{14}{x}=- 4x^4+\frac{14}{x}$, since it has a variable $x$ in the denominator, it is not a polynomial.

Answer:

$9x^4 - x^3-\frac{x}{5}$