To determine which polynomials have the same complex zeros, we need to examine the zeros of each polynomial given in the question.

The polynomials provided are:

1. $f(x) = x^4 + x^3 + 2x^2 + 4x - 8$
2. $g(x) = 2x^3 + 2x^2 + 8x + 8$
3. $h(x) = x^2 + 1$
4. $j(x) = x^2(x - 2i)(x + 2i)$
5. A cubic polynomial $k(x)$ such that $\frac{k(x)}{x + 1} = x^2 + 4$.

Let's analyze each one:

Step 1: Identify Potential Complex Zeros

1. For $h(x) = x^2 + 1$:

   • This polynomial has roots $x = \pm i$, which are purely imaginary complex zeros.

2. For $j(x) = x^2(x - 2i)(x + 2i)$:

   • This polynomial includes the factors $(x - 2i)$ and $(x + 2i)$, so it has complex roots $x = \pm 2i$.

3. For $k(x)$:

   • To find $k(x)$, note that $\frac{k(x)}{x + 1} = x^2 + 4$.
   • Multiply both sides by $x + 1$: $k(x) = (x + 1)(x^2 + 4)$.
   • Expanding this, we get: $k(x) = x^3 + x^2 + 4x + 4$
   • The equation $x^2 + 4 = 0$ has roots $x = \pm 2i$, so $k(x)$ has complex roots $x = \pm 2i$.

Step 2: Compare for Matching Complex Zeros

• Polynomials with complex zeros $x = \pm i$: Only $h(x)$.
• Polynomials with complex zeros $x = \pm 2i$: Both $j(x)$ and $k(x)$.

Conclusion

The polynomials that have the same complex zeros are:

• $j(x)$ and $k(x)$, as they both have $x = \pm 2i$ as complex roots.

Would you like further details on these calculations?

Here are five additional questions that build on this analysis:

1. How do you find complex roots for a given polynomial in general?
2. What are the steps for factoring a polynomial to reveal its roots?
3. How does the presence of imaginary roots affect the graph of a polynomial?
4. Can a polynomial of odd degree have only complex roots?
5. How can we verify if two polynomials have identical roots without solving each polynomial?

Tip: When analyzing polynomials, always look for factors or transformations that simplify the polynomial into forms where you can easily identify roots.