Which one is a polynomial?

The correct answer is C: x² - 2x + √2. A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. This means that variables cannot be in the denominator, cannot have fractional exponents (like square roots of variables), and cannot have negative exponents.

• Option C: x² - 2x + √2 fits this definition perfectly. The exponents of x are 2 and 1 (both non-negative integers), and √2 is a constant coefficient, which is allowed. There are no variables in the denominator or under a radical.

Let's look at why the other options are not polynomials:

• Option A: 3x² + 1/x - 5 is not a polynomial because of the term 1/x. This can be rewritten as x⁻¹, which has a negative exponent, violating the rule for polynomial terms.
• Option B: 3x² - 4x² - x√x is not a polynomial because of the term x√x. This can be rewritten as x¹ * x^(1/2) = x^(3/2), which has a fractional exponent (3/2). Polynomials only allow whole number (non-negative integer) exponents.
• Option D: 3/(2x) * x + x + 8 is generally not considered a polynomial in its initial form. While it simplifies to 3/2 + x + 8 (which is x + 19/2), an expression is typically assessed as a polynomial based on the direct presentation of its terms. The term 3/(2x) initially places a variable in the denominator. Although the subsequent multiplication by x cancels it out, the presence of a variable in the denominator in any part of the expression often disqualifies it from being classified as a polynomial in its given form for such questions, especially when compared to a clearly structured polynomial like option C.