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The given polynomial is \( P(x) = x^3 + 11x^2 + 8x - 20 \). We need to find all the real zeros of this polynomial and then write it in factored form.

### Step 1: Try Rational Root Theorem
The Rational Root Theorem suggests that possible rational roots are factors of the constant term (-20) divided by factors of the leading coefficient (1). So the possible rational roots are:
\[
\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20
\]

### Step 2: Synthetic Division or Substitution
Let's check possible roots by substitution or synthetic division.

#### Test \( x = 1 \):
\[
P(1) = 1^3 + 11(1^2) + 8(1) - 20 = 1 + 11 + 8 - 20 = 0
\]
So, \( x = 1 \) is a root.

### Step 3: Polynomial Division
Now, divide \( P(x) \) by \( (x - 1) \) using synthetic or long division.

Performing synthetic division of \( P(x) = x^3 + 11x^2 + 8x - 20 \) by \( (x - 1) \):

\[
\begin{array}{r|rrrr}
1 & 1 & 11 & 8 & -20 \\
  &   & 1  & 12 & 20 \\
\hline
  & 1 & 12 & 20 & 0 \\
\end{array}
\]

The quotient is \( x^2 + 12x + 20 \).

### Step 4: Solve the Quadratic
Now, solve \( x^2 + 12x + 20 = 0 \) using the quadratic formula:
\[
x = \frac{-12 \pm \sqrt{12^2 - 4(1)(20)}}{2(1)} = \frac{-12 \pm \sqrt{144 - 80}}{2} = \frac{-12 \pm \sqrt{64}}{2} = \frac{-12 \pm 8}{2}
\]
Thus, \( x = \frac{-12 + 8}{2} = -2 \) or \( x = \frac{-12 - 8}{2} = -10 \).

### Step 5: Write the Factored Form
The real zeros are \( x = 1, -2, -10 \). Therefore, the polynomial in factored form is:
\[
P(x) = (x - 1)(x + 2)(x + 10)
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. What does the Rational Root Theorem state, and how is it applied?
2. Can synthetic division be used for non-rational roots?
3. How do we perform long division of polynomials?
4. What is the quadratic formula and when do we use it?
5. What are the differences between real and complex zeros of a polynomial?

**Tip**: When checking for rational roots, always try simple numbers like \( \pm 1 \) first, as they are easier to compute!

Find all real zeros and the factored form of the polynomial P(x) = x^3 + 11x^2 + 8x - 20.

Learn how to find all real zeros and the factored form of the polynomial P(x) = x^3 + 11x^2 + 8x - 20. This problem explores polynomial division, the Rational Root Theorem, and solving quadratics to find zeros.

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