To solve the cubic equation \( x^3 - 77x - 240 = 0 \), we can use methods such as factoring, synthetic division, or the cubic formula. Here's the detailed process:

1. **Find the possible rational roots:** According to the Rational Root Theorem, the possible rational roots of the polynomial equation \( x^3 - 77x - 240 = 0 \) are the factors of the constant term (-240) divided by the factors of the leading coefficient (1). Thus, the possible rational roots are:
   \[
   \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 8, \pm 10, \pm 12, \pm 15, \pm 16, \pm 20, \pm 24, \pm 30, \pm 40, \pm 48, \pm 60, \pm 80, \pm 120, \pm 240
   \]

2. **Test the possible roots:** We substitute these values into the equation to see if they yield zero.
   \[
   x = 5: \quad 5^3 - 77 \cdot 5 - 240 = 125 - 385 - 240 = 0
   \]
   So, \( x = 5 \) is a root.

3. **Factor the polynomial:** Once we find a root, we can factor the polynomial using synthetic division.
   \[
   \begin{array}{r|rrrr}
     5 & 1 & 0 & -77 & -240 \\
       &   & 5 & 25 & -260 \\
     \hline
       & 1 & 5 & -52 & 0 \\
   \end{array}
   \]
   So, the factor is \( (x - 5) \), and we get:
   \[
   x^3 - 77x - 240 = (x - 5)(x^2 + 5x - 52)
   \]

4. **Solve the quadratic equation:** Next, solve the quadratic equation \( x^2 + 5x - 52 = 0 \) using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 1 \), \( b = 5 \), and \( c = -52 \).
   \[
   x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-52)}}{2 \cdot 1}
   \]
   \[
   x = \frac{-5 \pm \sqrt{25 + 208}}{2}
   \]
   \[
   x = \frac{-5 \pm \sqrt{233}}{2}
   \]

So, the roots of the equation \( x^3 - 77x - 240 = 0 \) are:
   \[
   x = 5, \quad x = \frac{-5 + \sqrt{233}}{2}, \quad x = \frac{-5 - \sqrt{233}}{2}
   \]

Would you like more details on any step or have any questions?

Here are some related questions you might consider:
1. How can I verify the roots of a polynomial equation?
2. Can you explain synthetic division in more detail?
3. How do you apply the Rational Root Theorem to other polynomials?
4. What is the cubic formula and when is it used?
5. How do you factor polynomials using the quadratic formula?

**Tip:** When solving polynomial equations, always check for possible rational roots first, as they can simplify the problem significantly.

Learn how to solve the cubic equation x^3 - 77x - 240 = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomial equations, the Rational Root Theorem, and the quadratic formula. Suitable for students in Grades 10-12.

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