https://file.aiquickdraw.com/mathbot/file/17254633417071a2vll8s.jpg

The problem presented in the image is:

Given \( f(x) = x^3 - 3x^2 + 2x - 5 \), what is \( f(2) - f(3) \)?

Let's solve this step by step.

1. **Find \( f(2) \):**

\[
f(2) = 2^3 - 3(2)^2 + 2(2) - 5
\]
\[
f(2) = 8 - 12 + 4 - 5 = -5
\]

2. **Find \( f(3) \):**

\[
f(3) = 3^3 - 3(3)^2 + 2(3) - 5
\]
\[
f(3) = 27 - 27 + 6 - 5 = 1
\]

3. **Calculate \( f(2) - f(3) \):**

\[
f(2) - f(3) = -5 - 1 = -6
\]

Therefore, the correct answer is \(-6\), which corresponds to the option with \(-6\).

Would you like further details or have any questions? Here are some related questions to consider:

1. What is the significance of the cubic term in \( f(x) \)?
2. How does changing the coefficient of the quadratic term affect the graph of \( f(x) \)?
3. What would \( f(x) - f(y) \) represent geometrically?
4. How can the remainder theorem be applied to evaluate \( f(2) \) and \( f(3) \)?
5. What happens to \( f(x) \) as \( x \) approaches infinity?

**Tip:** When dealing with polynomial functions, always carefully substitute values step-by-step to avoid mistakes.

Given f(x) = x^3 - 3x^2 + 2x - 5, what is f(2) - f(3)?

Learn how to evaluate the polynomial function f(x) = x^3 - 3x^2 + 2x - 5 by calculating f(2) and f(3), then finding the difference. This problem covers key algebraic concepts, suitable for students in Grades 9-11.

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