The polynomial provided is:

$P(x) = x^4 - 3x^3 - 27x^2 - 3x - 28$

We need to find all the zeros of this polynomial. To do this, we'll first check if the polynomial can be factored or if there are any rational roots by using the Rational Root Theorem. This theorem suggests that any rational root must be a factor of the constant term (-28) divided by a factor of the leading coefficient (1). Hence, the possible rational roots are:

$\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28$

I'll now check these potential roots and proceed to factor or solve the polynomial step by step.The polynomial $P(x) = x^4 - 3x^3 - 27x^2 - 3x - 28$ has the following zeros:

$x = -4, 7, i, -i$

These include both real and complex roots:

• Real roots: $-4$ and $7$
• Complex roots: $i$ and $-i$ (where $i$ is the imaginary unit).

Do you want further details on how these roots were determined or have any other questions?

Here are some related questions:

1. How does the Rational Root Theorem help in solving polynomials?
2. What is the significance of complex roots in a polynomial?
3. How can we factor polynomials once we know their roots?
4. What are possible methods to verify the roots of a polynomial?
5. How can complex conjugate pairs impact the symmetry of the polynomial?

Tip: Always check for both real and complex solutions when dealing with higher-degree polynomials.