Which expression is a polynomial in (x), while (z) acts like a coefficient?

Correct answer: \(z^2x^4+3zx^2-8\)

In \(z^2x^4+3zx^2-8\), the powers of \(x\) are \(4\), \(2\), and \(0\). All are non-negative integers, so it is a polynomial in \(x\). The expressions \(z^2\) and \(3z\) act as coefficients of powers of \(x\). In option A, \(x\) has exponent \(-2\); in option C, \(x\) is in the denominator, so its exponent is \(-1\); and in option D, the exponent of \(x\) is \(\frac12\). Hence, these are not polynomials in \(x\). Exam tip: when testing for a polynomial in \(x\), check the exponents of \(x\); other letters may be coefficients.