To find the complex zeros of a polynomial function, we first need to understand the polynomial itself. Although you didn’t provide a specific polynomial, I will explain the general process.
Let’s say our polynomial is f(x) = x3 + 3x2 + 3x + 1. To find the zeros, we would typically set the polynomial equal to zero:
f(x) = 0
Next, we can look for rational roots using the Rational Root Theorem or try to factor the polynomial if it’s simple enough. If we don’t find rational roots, this might suggest that the zeros are either irrational or complex.
For this polynomial, we can use synthetic division or polynomial long division to factor it, or apply techniques like completing the square or applying the quadratic formula if it’s a quadratic polynomial.
Assuming we find a factorable form, let’s say we find that:
f(x) = (x + 1)(x2 + 2)
We can set each factor equal to zero. The first factor gives us:
x + 1 = 0 ⟹ x = -1
The second factor involves a quadratic. To find its zeros, we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
For the quadratic x2 + 2 = 0, we have:
a = 1, b = 0, c = 2
Calculating the discriminant:
Δ = 0² – 4(1)(2) = -8
Since the discriminant is negative, this indicates that the roots are complex:
x = (0 ± √(-8)) / 2(1) = ± √(8)i / 2 = ± 2√2i / 2 = ± √2i
Thus, the zeros of our polynomial function are:
-1, √2i, -√2i
In factored form, we can write:
f(x) = (x + 1)(x – √2i)(x + √2i)
This gives us both the real and complex roots in a clear factorized representation. Remember, the specific polynomial you deal with may result in different steps or outcomes, but the process generally remains the same.