Question

algebra: concepts and connections - plc introduction to polynomials which algebraic expression is a trinomial? options: $4x^3 + x^2 - \frac{1}{x}$, $x^6 - x + \sqrt{6}$, $x^3 + x^2 - \sqrt{x}$, $2x^3 - x^2$

Explanation:

Step1: Define trinomial polynomial

A trinomial is a polynomial with exactly 3 terms, where each term has non-negative integer exponents on the variable.

Step2: Analyze first expression

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

Step3: Analyze second expression

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

Step4: Analyze third expression

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

Step5: Analyze fourth expression

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

Answer:

$x^6 - x + \sqrt{6}$