Question

determine the number of zeros of the polynomial function.\\( f(x) = x^4 - 5x \\)\\( \underline{\quad} \\) zero(s)\\( \text{enter an exact number.} \\)

Explanation:

Step1: Factor out common term

$f(x) = x(x^3 - 5)$

Step2: Set function equal to 0

$x(x^3 - 5) = 0$

Step3: Solve for x values

First root: $x = 0$
Second root: $x^3 - 5 = 0 \implies x = \sqrt[3]{5}$
By the Fundamental Theorem of Algebra, a degree-4 polynomial has 4 total zeros (counting complex and multiplicities). The real zeros are 0 and $\sqrt[3]{5}$; the other two are complex conjugate zeros from the cubic factor's quadratic component after factoring $x^3-5=(x-\sqrt[3]{5})(x^2+\sqrt[3]{5}x+\sqrt[3]{25})$, which has two complex roots via quadratic formula.

Answer:

4