To find the zeros of the polynomial function $f(x) = x^5 - 5x^3 - 36x$, we can start by factoring out common terms and solving for $x$.

Step-by-Step Solution

1. Factor out $x$: $f(x) = x(x^4 - 5x^2 - 36)$ So, one of the zeros is $x = 0$.

2. Rewrite the remaining factor: We now have: $f(x) = x(x^4 - 5x^2 - 36)$ Notice that $x^4 - 5x^2 - 36$ is a quadratic in terms of $x^2$. Let $u = x^2$, then: $u^2 - 5u - 36 = 0$

3. Solve the quadratic equation $u^2 - 5u - 36 = 0$: Using the quadratic formula: $u = \frac{5 \pm \sqrt{25 + 144}}{2} = \frac{5 \pm \sqrt{169}}{2} = \frac{5 \pm 13}{2}$ This gives two solutions for $u$: $u = 9 \quad \text{and} \quad u = -4$

4. Return to $x$ by substituting back $u = x^2$:

   • For $u = 9$: $x^2 = 9 \Rightarrow x = 3 \quad \text{or} \quad x = -3$
   • For $u = -4$: $x^2 = -4 \Rightarrow x = 2i \quad \text{or} \quad x = -2i$

Final Answer

The zeros of the polynomial function $f(x) = x^5 - 5x^3 - 36x$ are: $x = 0, x = 3, x = -3, x = 2i, x = -2i$

Match with Options

From the options given, the correct answer is: D. $x = 0, x = -3, x = 3, x = 2i, x = -2i$

Would you like further explanation on any steps?

Here are some related questions for further practice:

1. How would you factor a polynomial of a higher degree using synthetic division?
2. What are complex roots and how are they represented on the complex plane?
3. How do you identify the multiplicity of a root?
4. Can all polynomial functions be factored over the real numbers?
5. What is the significance of the Fundamental Theorem of Algebra in finding polynomial roots?

Tip: When dealing with polynomial functions, always look for common factors to simplify the equation initially.