https://file.aiquickdraw.com/mathbot/file/1726266633163b8mmtduw.jpg

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?

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### 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!

Suppose f(x) = (x - 2)g(x), where g(x) is some polynomial. Find a solution: f(x) = 0. Suppose g(x) = 0 has solutions for x = 1, x = -7, and no others. Find the x-intercepts of the graph of y = f(x).

Learn how to solve f(x) = (x - 2)g(x) and find the x-intercepts when g(x) has solutions for x = 1 and x = -7. This problem explores key algebraic concepts, including polynomials and the zero product property. Suitable for students in Grades 9-11.

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