Question

which algebraic expression is a trinomial?
$x^3 + x^2 - \sqrt{x}$
$2x^3 - x^2$
$4x^3 + x^2 - \frac{1}{x}$
$x^6 - x + \sqrt{6}$

Explanation:

Step1: Define a trinomial

A trinomial is a polynomial with exactly 3 terms, where each term has non-negative integer exponents on the variable, and no variables in denominators or square roots.

Step2: Analyze Option 1

$x^3 + x^2 - \sqrt{x} = x^3 + x^2 - x^{\frac{1}{2}}$. The exponent $\frac{1}{2}$ is not a non-negative integer, so this is not a polynomial, hence not a trinomial.

Step3: Analyze Option 2

$2x^3 - x^2$ has only 2 terms, so it is a binomial, not a trinomial.

Step4: Analyze Option 3

$4x^3 + x^2 - \frac{1}{x} = 4x^3 + x^2 - x^{-1}$. The exponent $-1$ is negative, so this is not a polynomial, hence not a trinomial.

Step5: Analyze Option 4

$x^6 - x + \sqrt{6}$ has 3 terms: $x^6$, $-x$, and $\sqrt{6}$. All variable exponents are non-negative integers, so this is a trinomial.

Answer:

$\boldsymbol{x^6 - x + \sqrt{6}}$ (the last option)