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The exact values for all four zeros (real and complex) of the polynomial \( P(x) = 6x^4 + 23x^3 + 14x^2 - 53x - 60 \) are:

1. \( x = -\frac{4}{3} \) (real)
2. \( x = \frac{3}{2} \) (real)
3. \( x = -2 - i \) (complex)
4. \( x = -2 + i \) (complex)

### Explanation of How the Zeros Were Found:
1. **Factoring and Solving**: The polynomial was factored, revealing two real roots and two complex roots.
2. **Complex Conjugates**: The complex zeros appear as conjugates because the coefficients of the polynomial are real, which guarantees that any non-real zeros will occur in conjugate pairs.

Would you like more detailed steps or have further questions?

Here are 5 related questions:
1. How do we factor polynomials of degree 4?
2. What is the significance of complex conjugates in polynomial equations?
3. Can synthetic division be used to verify these roots?
4. How does Descartes' Rule of Signs apply to finding real roots?
5. How can we graph the polynomial to visualize the zeros?

**Tip**: Always verify polynomial roots by substituting them back into the original equation.

Give exact values for all four zeros (real and complex) of the polynomial. In your work, explain how you found each zero.
P(x) = 6x^4 + 23x^3 + 14x^2 - 53x - 60

Learn how to find the exact values of all four zeros (real and complex) of the polynomial P(x) = 6x^4 + 23x^3 + 14x^2 - 53x - 60. This step-by-step guide covers factoring polynomials, using the quadratic formula, and understanding complex conjugates. Suitable for high school students in Grades 10-12.

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