A student says that \(P(x)=4x^3-5x^{-1}+2\) is a polynomial because all its coefficients are real. Which is the correct analysis of the student's error?

Correct answer: It is not a polynomial because one exponent of \(x\) is \(-1\).

In a polynomial, the exponents of a variable must be non-negative integers such as \(0,1,2,\ldots\). Here, \(x^{-1}=\frac{1}{x}\), so \(-5x^{-1}\) cannot be a polynomial term. Having real coefficients is not sufficient; the exponents must also be non-negative integers. Exam tip: before identifying an expression as a polynomial, check every exponent of the variable.