1. **Problem Statement:** We are given a polynomial function $f(x)$ with the following features: - Degree is even - Leading coefficient is negative - There are 4 distinct real zeros - There are 3 relative extrema 2. **Understanding the Problem:** - The degree being even means the ends of the graph go in the same direction. - A negative leading coefficient means the ends of the graph go downwards. - 4 distinct real zeros means the graph crosses the x-axis at 4 different points. - 3 relative extrema means there are 3 peaks or valleys where the graph changes direction. 3. **General Form of Polynomial:** A polynomial with 4 real zeros $r_1, r_2, r_3, r_4$ can be written as: $$f(x) = a(x - r_1)(x - r_2)(x - r_3)(x - r_4)$$ where $a$ is the leading coefficient. 4. **Degree and Leading Coefficient:** - Since there are 4 zeros, the degree is 4 (which is even). - The leading coefficient $a$ is negative. 5. **Example Polynomial:** Choose zeros at $-3, -1, 1, 3$ for simplicity: $$f(x) = -1(x + 3)(x + 1)(x - 1)(x - 3)$$ 6. **Simplify:** First, group factors: $$(x + 3)(x - 3) = x^2 - 9$$ $$(x + 1)(x - 1) = x^2 - 1$$ So, $$f(x) = -1(x^2 - 9)(x^2 - 1)$$ 7. **Expand:** $$f(x) = -1(x^4 - x^2 - 9x^2 + 9) = -1(x^4 - 10x^2 + 9)$$ $$f(x) = -x^4 + 10x^2 - 9$$ 8. **Summary:** - Degree: 4 (even) - Leading coefficient: $-1$ (negative) - 4 distinct real zeros at $-3, -1, 1, 3$ - 3 relative extrema (peaks and valleys) as expected for a quartic with this shape **Final answer:** $$f(x) = -x^4 + 10x^2 - 9$$