All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

*P*(*x*) = *x*3 − 6*x*2 + 12*x* − 8

Given the polynomial \( P(x) = x^3 - 6x^2 + 12x - 8 \), we need to find all the real zeros. Since all the real zeros are integers, we can apply the Rational Root Theorem and synthetic division or direct factoring.

### Step 1: Check for possible rational roots
The Rational Root Theorem suggests that any rational root is a factor of the constant term divided by a factor of the leading coefficient. For the given polynomial:

- The constant term is \(-8\).
- The leading coefficient (of \( x^3 \)) is \( 1 \).

The possible rational roots are the factors of \(-8\): \(\pm 1, \pm 2, \pm 4, \pm 8\).

### Step 2: Test possible roots
Let's test the possible roots by direct substitution or using synthetic division.

1. **Test \( x = 2 \):**
   \[
   P(2) = (2)^3 - 6(2)^2 + 12(2) - 8 = 8 - 24 + 24 - 8 = 0
   \]
   Since \( P(2) = 0 \), \( x = 2 \) is a root.

### Step 3: Factor the polynomial
Since \( x = 2 \) is a root, we can factor \( P(x) \) as:
\[
P(x) = (x - 2)Q(x)
\]
where \( Q(x) \) is a quadratic polynomial.

To find \( Q(x) \), perform synthetic division of \( P(x) \) by \( x - 2 \):

\[
\begin{array}{r|rrr}
  2 & 1 & -6 & 12 & -8  \\
    &   & 2  & -8 & 8  \\
  \hline
    & 1 & -4 & 4  & 0  \\
\end{array}
\]

The quotient is \( Q(x) = x^2 - 4x + 4 \).

Thus, we can factor the polynomial as:
\[
P(x) = (x - 2)(x^2 - 4x + 4)
\]

### Step 4: Factor the quadratic
Notice that the quadratic \( x^2 - 4x + 4 \) is a perfect square:
\[
x^2 - 4x + 4 = (x - 2)^2
\]

So the complete factorization of \( P(x) \) is:
\[
P(x) = (x - 2)(x - 2)(x - 2) = (x - 2)^3
\]

### Step 5: Find all zeros
Since \( P(x) = (x - 2)^3 \), the only root of the polynomial is \( x = 2 \), but with multiplicity 3.

### Final Answer:
The zeros of the polynomial \( P(x) = x^3 - 6x^2 + 12x - 8 \) are \( \boxed{2, 2, 2} \).

Would you like more details, or do you have any questions?

Here are five related questions:
1. What is the multiplicity of a root and how does it affect the graph of a polynomial?
2. How would you find the zeros of a polynomial if they are not all integers?
3. What is the Rational Root Theorem and how is it applied?
4. Can you explain the process of synthetic division in more detail?
5. How would the graph of \( P(x) = x^3 - 6x^2 + 12x - 8 \) look based on the zeros?

**Tip:** Knowing how to factor polynomials efficiently can greatly simplify solving for their roots.

Learn how to find all integer zeros of the polynomial x^3 - 6x^2 + 12x - 8 using the Rational Root Theorem and synthetic division. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation