We are tasked with finding the real zeros of the polynomial and writing it in factored form.

Given polynomial: $P(x) = x^4 - 2x^3 - 8x^2 + 18x - 9$

Step 1: Use the Rational Root Theorem

The Rational Root Theorem suggests that possible rational roots are factors of the constant term (in this case, $-9$) divided by factors of the leading coefficient (in this case, $1$).

The factors of $-9$ are: $\pm 1, \pm 3, \pm 9$

Now, we will check these potential zeros by substituting them into the polynomial.

Step 2: Test possible rational roots

Test $x = 1$:

$P(1) = (1)^4 - 2(1)^3 - 8(1)^2 + 18(1) - 9 = 1 - 2 - 8 + 18 - 9 = 0$ So, $x = 1$ is a root.

Test $x = -1$:

$P(-1) = (-1)^4 - 2(-1)^3 - 8(-1)^2 + 18(-1) - 9 = 1 + 2 - 8 - 18 - 9 = -32$ $x = -1$ is not a root.

Test $x = 3$:

$P(3) = (3)^4 - 2(3)^3 - 8(3)^2 + 18(3) - 9 = 81 - 54 - 72 + 54 - 9 = 0$ So, $x = 3$ is a root.

Test $x = -3$:

$P(-3) = (-3)^4 - 2(-3)^3 - 8(-3)^2 + 18(-3) - 9 = 81 + 54 - 72 - 54 - 9 = 0$ So, $x = -3$ is a root.

Step 3: Polynomial Division

Now that we know $x = 1$, $x = 3$, and $x = -3$ are roots, we can divide the polynomial by $(x - 1)(x - 3)(x + 3)$.

First, factor $(x^2 - 9)$ from $(x - 3)(x + 3)$: $P(x) = (x - 1)(x^2 - 9)Q(x)$

Now, divide the polynomial $P(x)$ by $(x^3 - x^2 - 9x + 9)$ using synthetic or long division, which yields $x - 1$ as the quotient.

Thus, the polynomial can be factored as: $P(x) = (x - 1)^2(x - 3)(x + 3)$

Final Answer:

The zeros are: $x = 1 \text{ (with multiplicity 2)}, 3, -3$

The factored form of the polynomial is: $P(x) = (x - 1)^2(x - 3)(x + 3)$

Let me know if you need more details or clarifications!

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Here are 5 related questions you could explore:

1. What happens if one of the roots has a higher multiplicity?
2. How does the Rational Root Theorem help narrow down possible solutions?
3. Can all polynomials be factored into real roots?
4. How do you confirm the multiplicity of a root?
5. What other methods can be used to find the roots of a polynomial?

Tip: When checking for rational roots, always test the smallest factors first to save time.