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The given polynomial is:

\[
P(x) = x^5 - x^4 - 6x^3 + 14x^2 - 11x + 3
\]

We need to find all real zeros of \( P(x) \). To do this, we'll start by applying methods like factoring (if possible), or we may need to use numerical or graphical methods to approximate the roots.

Let me calculate the roots of the polynomial.The real zeros of the polynomial \( P(x) = x^5 - x^4 - 6x^3 + 14x^2 - 11x + 3 \) are approximately:

\[
x = -3, \, 1, \, 1
\]

Thus, the real zeros are: \(-3, 1, 1\).

Would you like any further details or clarifications?

Here are some related questions for further practice:

1. How can you verify if a given value is a zero of a polynomial?
2. What are the complex zeros of the polynomial \( P(x) \)?
3. How do multiplicities of zeros affect the graph of a polynomial?
4. Can this polynomial be factored further into smaller degree polynomials?
5. How would the graph of this polynomial look based on its zeros and degree?

**Tip**: If a root repeats, it means the polynomial touches but does not cross the x-axis at that point.

Find all the real zeros of the polynomial P(x) = x^5 - x^4 - 6x^3 + 14x^2 - 11x + 3.

Solve the polynomial P(x) = x^5 - x^4 - 6x^3 + 14x^2 - 11x + 3 and find all its real zeros. This solution applies factorization, numerical methods, and graphical analysis. Suitable for high school students studying algebra and polynomial functions.

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