Which polynomial is given in (z) instead of (x)?

Correct answer: (4z^3-z+8)

A polynomial in a variable is written using non-negative whole-number powers of that variable. The letter used as the variable may be \(x\), \(z\), or another symbol; the definition does not require the variable to be called x. In \(4z^3-z+8\), the variable is z, and the powers of z are 3, 1, and 0. All are non-negative integers, so this is a polynomial in z.

Option A and option C are polynomials in x. Option D is not a polynomial in the usual school definition because it contains \(z^{-1}\), a negative power, and it also involves z. The question asks which polynomial is given in z instead of x, so option B is the intended and correct choice. Its three terms and allowed powers satisfy the polynomial definition. The supplied answer and explanation are accurate.