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We are given the polynomial \( P(x) = x^4 - 3x^3 + 27x - 81 \) and asked to factor it and find its zeros.

### Step 1: Factor the polynomial

We can begin by testing for possible rational roots using the Rational Root Theorem, which suggests testing factors of the constant term (81) divided by factors of the leading coefficient (1). These possible rational roots are \( \pm 1, \pm 3, \pm 9, \pm 27, \pm 81 \).

Let's test these values to see if any are zeros.

### Step 2: Synthetic division or direct substitution
We can start with synthetic division or substitution to test these roots. I will compute this step now.From the substitution, we find that the polynomial evaluates to 0 at \( x = 3 \) and \( x = -3 \). This means that both \( x = 3 \) and \( x = -3 \) are roots of the polynomial.

### Step 3: Factor the polynomial
Since \( x = 3 \) and \( x = -3 \) are roots, we know that \( (x - 3) \) and \( (x + 3) \) are factors. Next, we divide the polynomial \( P(x) \) by \( (x - 3)(x + 3) = x^2 - 9 \) to find the remaining factor.

I will perform the division now to factor the polynomial fully.After dividing \( P(x) \) by \( (x^2 - 9) \), the quotient is \( x^2 - 3x + 9 \), and the remainder is 0. Therefore, the complete factorization of the polynomial is:

\[
P(x) = (x - 3)(x + 3)(x^2 - 3x + 9)
\]

### Step 4: Solve for the zeros
To find the zeros, set each factor equal to zero:

1. \( x - 3 = 0 \) gives \( x = 3 \),
2. \( x + 3 = 0 \) gives \( x = -3 \),
3. \( x^2 - 3x + 9 = 0 \).

Let's solve the quadratic \( x^2 - 3x + 9 = 0 \) using the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(9)}}{2(1)}
\]

This gives:

\[
x = \frac{3 \pm \sqrt{9 - 36}}{2} = \frac{3 \pm \sqrt{-27}}{2} = \frac{3 \pm 3i\sqrt{3}}{2}
\]

Thus, the complex roots are:

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

### Final Answer:
The zeros of \( P(x) \) are:
- Real roots: \( x = 3 \), \( x = -3 \),
- Complex roots: \( x = \frac{3}{2} + \frac{3i\sqrt{3}}{2} \) and \( x = \frac{3}{2} - \frac{3i\sqrt{3}}{2} \).

Would you like a more detailed explanation of any of the steps?

### Related Questions:
1. How do you factor other types of quartic polynomials?
2. What is the Rational Root Theorem and how is it applied?
3. Can you solve the quadratic equation \( x^2 - 3x + 9 = 0 \) step by step?
4. What are complex roots, and how do they occur in polynomials?
5. How do synthetic division and polynomial division differ in factoring?

### Tip:
When factoring polynomials, always test small integer roots first. This helps simplify the problem quickly.

Factor the polynomial P(x) = x^4 - 3x^3 + 27x - 81 and use the factored form to find the zeros.

This guide explains how to factor the polynomial P(x) = x^4 - 3x^3 + 27x - 81 and find its zeros. The problem explores factoring polynomials, applying the Rational Root Theorem, and solving for complex roots using the quadratic formula. Suitable for high school students in Grades 10-12.

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