To solve the cubic equation $X^3 + X^2 + X + 1 = 0$, let's use a step-by-step approach:

Step 1: Factor the Equation

We can try factoring by grouping or testing some values of $X$ to see if they satisfy the equation.

First, let's test $X = -1$:

$(-1)^3 + (-1)^2 + (-1) + 1 = -1 + 1 - 1 + 1 = 0$

Since $X = -1$ is a root of the equation, we can now factor the cubic polynomial by dividing it by $(X + 1)$.

Step 2: Divide the Polynomial by $X + 1$

We can perform synthetic division or long division:

$\frac{X^3 + X^2 + X + 1}{X + 1} = X^2 + 1$

Step 3: Solve the Quadratic $X^2 + 1 = 0$

Now we solve the quadratic equation:

$X^2 + 1 = 0$ $X^2 = -1$ $X = \pm i$

Thus, the solutions to the equation are: $X = -1, \quad X = i, \quad X = -i$

Final Answer:

The solutions are: $X = -1, \quad X = i, \quad X = -i$

Do you have any questions or want more details on any of the steps?

Here are some related questions to expand the topic:

1. How does synthetic division work for polynomial equations?
2. What is the general solution for quadratic equations with complex roots?
3. How can we graph complex roots in the complex plane?
4. How do we solve cubic equations that don't have an obvious factor?
5. What are the properties of the complex unit $i$?

Tip: When solving higher-degree polynomials, always check for simple roots like $X = 0$, $X = 1$, or $X = -1$ to simplify the equation.