Let's break down the problem.

1. We are given the function:

$f(x) = (x - 2)g(x)$ where $g(x)$ is some polynomial.

2. The first part asks to find a solution to $f(x) = 0$.

For $f(x) = 0$, we know: $(x - 2)g(x) = 0$ This equation is satisfied when either:

• $x - 2 = 0$, which gives $x = 2$, or
• $g(x) = 0$.

Since $g(x) = 0$ has solutions at $x = 1$ and $x = -7$ (and no others), the solutions to $f(x) = 0$ are: $x = 2, 1, -7$

Thus, the x-intercepts of the graph of $f(x)$ are $x = 2, 1, -7$.

Summary of Results:

• One solution to $f(x) = 0$ is $x = 2$.
• The x-intercepts of the graph of $y = f(x)$ are $x = 2, 1, -7$.

Would you like further clarification on any part of the solution?

5 Related Questions:

1. What happens to $f(x)$ when $x \neq 1, -7, 2$?
2. How would the graph of $f(x)$ look based on these intercepts?
3. What is the behavior of $g(x)$ as $x \to \infty$?
4. Could $g(x)$ be factored further if more information about it was given?
5. How would the multiplicity of roots affect the shape of the graph?

Tip:

When solving polynomial equations, always consider the factors and how each one could individually reach zero!