15*x*4 – 15*x*3 + 15*x* – 15 = 0

The given equation is:

\[
15x^4 - 15x^3 + 15x - 15 = 0
\]

Let's simplify and solve the equation step by step.

### Step 1: Factor out the common factor

Notice that each term on the left-hand side has a common factor of \( 15 \). We can factor this out:

\[
15(x^4 - x^3 + x - 1) = 0
\]

Since \( 15 \neq 0 \), the equation simplifies to:

\[
x^4 - x^3 + x - 1 = 0
\]

### Step 2: Look for potential rational roots

We can check for possible rational roots using the Rational Root Theorem, which suggests testing factors of the constant term (in this case, \(-1\)) over factors of the leading coefficient (which is \(1\)). The possible rational roots are \( \pm 1 \).

Let's test these values:

1. **For \( x = 1 \):**

   \[
   1^4 - 1^3 + 1 - 1 = 1 - 1 + 1 - 1 = 0
   \]

   Since \( x = 1 \) is a root, \( (x - 1) \) is a factor of the polynomial.

### Step 3: Factor the polynomial using synthetic division

Now that we know \( x = 1 \) is a root, we can perform synthetic division to factor the polynomial:

\[
x^4 - x^3 + x - 1 \text{ divided by } (x - 1)
\]

Performing synthetic division:

\[
\begin{array}{r|rrrrr}
1 & 1 & -1 & 0 & 1 & -1 \\
  &   & 1  & 0 & 0 & 1 \\
\hline
  & 1 & 0  & 0 & 1 & 0 \\
\end{array}
\]

The quotient is:

\[
x^3 + 1
\]

So, the polynomial can be factored as:

\[
(x - 1)(x^3 + 1) = 0
\]

### Step 4: Factor the cubic expression further

The cubic expression \( x^3 + 1 \) can be factored using the sum of cubes formula:

\[
x^3 + 1 = (x + 1)(x^2 - x + 1)
\]

So the complete factorization of the original equation is:

\[
(x - 1)(x + 1)(x^2 - x + 1) = 0
\]

### Step 5: Solve for \( x \)

Set each factor to zero:

1. \( x - 1 = 0 \) gives \( x = 1 \)
2. \( x + 1 = 0 \) gives \( x = -1 \)
3. \( x^2 - x + 1 = 0 \)

For the quadratic equation \( x^2 - x + 1 = 0 \), the discriminant \( \Delta = b^2 - 4ac \) is:

\[
\Delta = (-1)^2 - 4(1)(1) = 1 - 4 = -3
\]

Since the discriminant is negative, the roots are complex:

\[
x = \frac{1 \pm \sqrt{-3}}{2} = \frac{1 \pm \sqrt{3}i}{2}
\]

### Conclusion

The solutions to the equation are:

\[
\boxed{x = 1, \, x = -1, \, x = \frac{1 + \sqrt{3}i}{2}, \, x = \frac{1 - \sqrt{3}i}{2}}
\]

Would you like more details on any part of this solution?

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Here are five questions you might want to explore next:

1. How do you perform synthetic division with polynomials?
2. What is the Rational Root Theorem and how is it used?
3. How do you factor the sum or difference of cubes?
4. How do you solve quadratic equations with complex roots?
5. What are some examples of polynomials that have both real and complex roots?

**Tip:** When factoring polynomials, always check if the roots are real or complex by evaluating the discriminant, especially when dealing with quadratic factors.

Learn how to solve the polynomial equation 15x^4 - 15x^3 + 15x - 15 = 0 step-by-step with detailed explanations. This problem involves factoring polynomials, finding rational roots, and understanding complex number solutions. Suitable for high school students.

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