To solve the cubic equation \(x^3 + 7x + 8 = 0\), let's first try factoring or using numerical methods like the Rational Root Theorem or synthetic division, as there are no simple factoring methods for most cubic equations.

### Step 1: Try Rational Root Theorem
The Rational Root Theorem suggests that any rational solution to the equation must be a divisor of the constant term divided by the leading coefficient. Here, the constant term is 8, and the leading coefficient is 1.

Possible rational roots are the divisors of 8: \( \pm 1, \pm 2, \pm 4, \pm 8 \).

### Step 2: Test the possible roots

- Test \(x = -1\):
  \[
  (-1)^3 + 7(-1) + 8 = -1 - 7 + 8 = 0
  \]
  So, \(x = -1\) is a root.

### Step 3: Perform synthetic division
Now that we know \(x = -1\) is a root, we can divide the cubic polynomial by \(x + 1\) to reduce the degree of the polynomial.

Performing synthetic division of \(x^3 + 7x + 8\) by \(x + 1\) gives the quotient:
\[
x^2 - x + 8
\]

### Step 4: Solve the quadratic equation
Now, solve \(x^2 - x + 8 = 0\) using the quadratic formula:
\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(8)}}{2(1)} = \frac{1 \pm \sqrt{1 - 32}}{2} = \frac{1 \pm \sqrt{-31}}{2}
\]
\[
x = \frac{1 \pm \sqrt{31}i}{2}
\]

### Final solution:
The roots of the cubic equation are:
\[
x_1 = -1, \quad x_2 = \frac{1 + \sqrt{31}i}{2}, \quad x_3 = \frac{1 - \sqrt{31}i}{2}
\]

Would you like further details on any of the steps? Here are 5 related questions to explore:

1. How does the Rational Root Theorem work in general?
2. Can synthetic division be used for higher-degree polynomials?
3. What are imaginary numbers, and how do they relate to real numbers?
4. How can we visually interpret the roots of cubic equations on a graph?
5. How do we derive and apply the quadratic formula?

**Tip:** When solving cubic equations, always check for a rational root first, as it can simplify the process immensely.

Explore how to solve the cubic equation x^3 + 7x + 8 = 0 with a detailed step-by-step approach. Learn about using the Rational Root Theorem, synthetic division, and solving complex roots through the quadratic formula. Suitable for students in Grades 10-12.

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