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The problem requires finding the real solutions for the polynomial equation:

\[
x^3 - 7x^2 + 6x + 20 = 0
\]

To solve this graphically using a graphing calculator:
1. Enter the function \( f(x) = x^3 - 7x^2 + 6x + 20 \) into the graphing calculator.
2. Plot the function and observe where the graph crosses the x-axis. These points are the real solutions for the equation.
3. Zoom in on the points where the curve intersects the x-axis to get a more accurate reading to two decimal places.

If you'd like, I can also solve this analytically or describe how the function behaves. Let me know if you'd like further details or if you want a solution without a graphing calculator.

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Do you need more details on this? Here are some additional questions that might help:
1. What are the real and complex solutions for this cubic equation?
2. How can you use synthetic division to factor the polynomial?
3. How do the turning points and inflection points affect the graph of this polynomial?
4. What does the behavior of \( f(x) \) as \( x \to \pm \infty \) tell you about the polynomial’s end behavior?
5. Can this equation be solved using the Rational Root Theorem?

**Tip:** For graphing cubic polynomials, it's helpful to analyze the critical points (where the derivative is zero) to better understand the shape of the graph.

Use your graphing calculator to solve the equation graphically for all real solutions: x^3 - 7x^2 + 6x + 20 = 0.

Learn how to solve the cubic polynomial equation x^3 - 7x^2 + 6x + 20 = 0 using a graphing calculator. This problem explores algebraic concepts and graphing techniques, providing a step-by-step guide for finding the real solutions. Suitable for students in Grades 10-12.

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