To factor the polynomial $P(x) = 2x^3 - x^2 - 18x + 9$ and find its real zeros, let's go through the process step-by-step.

Step 1: Check for Possible Rational Roots

By the Rational Root Theorem, possible rational roots are factors of the constant term (9) divided by factors of the leading coefficient (2). Thus, possible roots are: $\pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2}$

Step 2: Synthetic Division or Substitution

We can test these values to see if any are roots of $P(x) = 0$. By trying some values, let’s start with $x = 3$.

Substitute $x = 3$:

$P(3) = 2(3)^3 - (3)^2 - 18(3) + 9 = 2 \cdot 27 - 9 - 54 + 9 = 54 - 9 - 54 + 9 = 0$ So, $x = 3$ is a root.

Step 3: Factor Out $x - 3$

Since $x = 3$ is a root, $x - 3$ is a factor. Now, divide $P(x)$ by $x - 3$ using synthetic division or polynomial division.

Synthetic Division:

1. Coefficients of $P(x) = 2x^3 - x^2 - 18x + 9$: $2, -1, -18, 9$.
2. Perform synthetic division with root 3: 3 & 2 & -1 & -18 & 9 \\ & & 6 & 15 & -9 \\ \hline & 2 & 5 & -3 & 0 \\ \end{array}$$ The quotient is $$2x^2 + 5x - 3$$.

Step 4: Factor the Quotient

Now, we need to factor $2x^2 + 5x - 3$: $2x^2 + 5x - 3 = (2x - 1)(x + 3)$

Step 5: Write the Factored Form of $P(x)$

So, the factored form of $P(x)$ is: $P(x) = (x - 3)(2x - 1)(x + 3)$

Step 6: Find the Real Zeros

Set each factor equal to zero to find the real roots:

1. $x - 3 = 0 \Rightarrow x = 3$
2. $2x - 1 = 0 \Rightarrow x = \frac{1}{2}$
3. $x + 3 = 0 \Rightarrow x = -3$

Answer

The real zeros of $P(x)$, including repetitions, are: $3, \frac{1}{2}, -3$

Would you like a more detailed explanation of any step? Here are some related questions to explore:

1. How can the Rational Root Theorem help in factoring polynomials?
2. What is synthetic division, and why is it useful?
3. What are other methods for factoring polynomials of higher degrees?
4. How can you verify the roots of a polynomial after finding them?
5. What is the significance of repeated roots in polynomial functions?

Tip: When factoring, always try simple roots first (like $\pm 1, \pm 3$) to simplify the process.