Final answer:
The polynomial function f(x) = x^4 - 6x has two real zeros, which are 0 and the cube root of 6. The other two zeros are complex, as polynomial functions have as many zeros as their degree, which in this case is four.
Step-by-step explanation:
The question involves finding the number of zeros of the given polynomial function f(x) = x4 - 6x. To find its zeros, we set the function equal to zero and factor where possible. The given polynomial can be rewritten as:
f(x) = x4 - 6x = x(x3 - 6)
Now, we can use the zero-product property which states that if a product of factors is zero, then at least one of the factors must be zero. This gives us two cases to consider:
- Case 1: x = 0
- Case 2: x3 - 6 = 0. This can be solved by adding 6 to both sides and then taking the cube root, which yields x = ∛6.
Thus, the polynomial f(x) has two real zeros: 0 and ∛6. The other two zeros are complex since all non-linear polynomials have as many zeros as their degree when complex numbers are considered.