https://file.aiquickdraw.com/mathbot/file/1725480472288gv6bnfo4.jpg

The function given in the image is:

\[
f(x) = 2.5x^4 + 3x^3 - 2.6x^2 - 5.1x - 5.6.
\]

To find all real zeros, we need to solve \(f(x) = 0\).

I will calculate the real zeros using numerical methods or factorization techniques.The real zeros of the function \( f(x) = 2.5x^4 + 3x^3 - 2.6x^2 - 5.1x - 5.6 \) are:

1. \( x = -1.6 \)
2. \( x = 1.4 \)

There are also two complex roots:
- \( x = -0.5 - 0.866i \)
- \( x = -0.5 + 0.866i \)

Would you like further details on how these zeros were found or have any other questions?

### Related Questions:
1. How do you use the Rational Root Theorem to find potential zeros?
2. What is a numerical method for finding roots, such as Newton's method?
3. How can you graph the function to verify these zeros visually?
4. What is the significance of complex roots in polynomial equations?
5. How can polynomial division help in finding roots?

**Tip:** When solving for real zeros, always check for potential rational roots first using factors of the constant term and the leading coefficient.

Learn how to find all real and complex zeros of the polynomial function f(x) = 2.5x^4 + 3x^3 - 2.6x^2 - 5.1x - 5.6. This problem involves polynomial equations, numerical methods, and complex number roots. Suitable for students in Grades 11-12.

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