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The polynomial given is:

\[
P(x) = x^4 - x^2 + 2x + 2
\]

To find the zeros of the polynomial, we need to solve for \( x \) when \( P(x) = 0 \). Let's go through the steps to find the roots.

### Step 1: Substitution to reduce degree
Let’s substitute \( y = x^2 \), so the polynomial becomes:

\[
y^2 - y + 2x + 2 = 0
\]

However, it seems that approach might not easily work here. Instead, we'll try solving the equation directly using factorization or numerical methods.

Let me calculate the roots of the polynomial.The roots of the polynomial \( P(x) = x^4 - x^2 + 2x + 2 \) are:

\[
1 + i, \quad 1 - i, \quad -1, \quad -1
\]

Thus, the zeros (including repetitions) are:

\[
1 + i, 1 - i, -1, -1
\]

Would you like more details on how these roots were computed or their verification? Let me know if you have any questions!

Here are 5 follow-up questions you may consider:
1. How can complex numbers affect the behavior of polynomial graphs?
2. What methods can be used to approximate complex roots of polynomials?
3. How do the multiplicities of roots affect the shape of the polynomial graph?
4. Could factoring have been a feasible method for solving this polynomial?
5. What real-world applications might involve solving polynomials with complex roots?

**Tip:** When solving polynomials, always check for patterns like factorization or substitution that may simplify the process before resorting to numerical solutions.

Find all zeros of the polynomial P(x) = x^4 - x^2 + 2x + 2

Learn how to find all the zeros of the polynomial P(x) = x^4 - x^2 + 2x + 2. This example covers concepts of algebra and complex numbers, using numerical methods to solve higher-degree polynomials.

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