To determine how many x-intercepts the function $f(x) = x(x - 2)^2(x^2 + 25)(x + 8)(x^2 - 5)$ has, we need to set $f(x) = 0$ and solve for $x$. X-intercepts occur where the function equals zero, i.e., where the factors of the function are equal to zero.

The factors of $f(x)$ are:

1. $x$
2. $(x - 2)^2$
3. $(x^2 + 25)$
4. $(x + 8)$
5. $(x^2 - 5)$

Let's solve for $x$ from each factor:

1. From $x = 0$: This gives one intercept at $x = 0$.

2. From $(x - 2)^2 = 0$: Solving this, $x = 2$. Since this factor is squared, $x = 2$ is a repeated root, meaning it only counts as one intercept because it just touches the x-axis.

3. From $(x^2 + 25) = 0$: Solving this: $x^2 = -25$ This has no real solutions (since you can't take the square root of a negative number), so it doesn't contribute any x-intercepts.

4. From $x + 8 = 0$: Solving this, $x = -8$. This gives one intercept at $x = -8$.

5. From $(x^2 - 5) = 0$: Solving this: $x^2 = 5 \quad \Rightarrow \quad x = \pm \sqrt{5}$ These are two real solutions, $x = \sqrt{5}$ and $x = -\sqrt{5}$, giving two more intercepts.

Summary of x-intercepts:

• $x = 0$
• $x = 2$ (only one intercept due to the squared factor)
• $x = -8$
• $x = \sqrt{5}$
• $x = -\sqrt{5}$

Conclusion:

There are 5 x-intercepts in total.

Would you like more details on any of these steps, or do you have any questions?

Here are 5 related questions to expand on this topic:

1. What is the significance of a double root for x-intercepts?
2. How does the multiplicity of roots affect the shape of the graph?
3. What is the geometric interpretation of complex roots for a polynomial function?
4. How can you find the y-intercept of this function?
5. How would adding another factor change the number of x-intercepts?

Tip: The degree of the polynomial gives insight into the number of possible real roots, but complex roots do not contribute to x-intercepts.