All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers using the appropriate multiplicities.)
P(x) = x4 − 2x3 − 8x2 + 18x − 9
x =
Write the polynomial in factored form.
P(x) =

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.

All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers using the appropriate multiplicities.) P(x) = x^4 − 2x^3 − 8x^2 + 18x − 9. Write the polynomial in factored form.

Discover how to find the real zeros of the polynomial P(x) = x^4 − 2x^3 − 8x^2 + 18x − 9 using the Rational Root Theorem and polynomial division. This solution provides step-by-step instructions for factoring the polynomial and identifying its real zeros. Ideal for students in Grades 10-12.

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