To find the zeros of the polynomial \( P(x) = x^3 - 8x^2 + 25x - 26 \), we will use a combination of the Rational Root Theorem and synthetic division.

### Step 1: Use the Rational Root Theorem
The Rational Root Theorem suggests that any rational root, if it exists, must be of the form \( \frac{p}{q} \), where:
- \( p \) is a factor of the constant term (-26)
- \( q \) is a factor of the leading coefficient (1)

The factors of -26 are: \( \pm 1, \pm 2, \pm 13, \pm 26 \)
The factors of 1 are: \( \pm 1 \)

Thus, the possible rational roots are: \( \pm 1, \pm 2, \pm 13, \pm 26 \).

### Step 2: Test potential roots
Let's test some possible roots using synthetic substitution.

#### Testing \( x = 1 \):

\[
\begin{array}{r|rrrr}
1 & 1 & -8 & 25 & -26 \\
  &   & 1  & -7 & 18  \\
\hline
  & 1 & -7 & 18 & -8
\end{array}
\]

Since the remainder is -8, \( x = 1 \) is not a root.

#### Testing \( x = 2 \):

\[
\begin{array}{r|rrrr}
2 & 1 & -8 & 25 & -26 \\
  &   & 2  & -12 & 26 \\
\hline
  & 1 & -6 & 13 & 0
\end{array}
\]

Since the remainder is 0, \( x = 2 \) is a root.

### Step 3: Factor the polynomial
Since \( x = 2 \) is a root, we can factor \( P(x) \) as:

\[
P(x) = (x - 2)(x^2 - 6x + 13)
\]

### Step 4: Solve the quadratic \( x^2 - 6x + 13 = 0 \)
Use the quadratic formula to find the remaining roots:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(13)}}{2(1)} = \frac{6 \pm \sqrt{36 - 52}}{2} = \frac{6 \pm \sqrt{-16}}{2} = \frac{6 \pm 4i}{2}
\]

This simplifies to:

\[
x = 3 \pm 2i
\]

### Final answer:
The zeros of \( P(x) = x^3 - 8x^2 + 25x - 26 \) are:

\[
x = 2, \quad x = 3 + 2i, \quad x = 3 - 2i
\]

Would you like further details on any part of the solution? Here are some related questions:

1. How do you use synthetic division for root finding?
2. What is the Rational Root Theorem and why is it useful?
3. How do you solve quadratic equations with complex roots?
4. What are imaginary and complex numbers?
5. How do cubic polynomials typically behave on a graph?

**Tip:** Always test smaller factors first when using the Rational Root Theorem to save time.

Discover how to find the zeros of the polynomial P(x) = x^3 - 8x^2 + 25x - 26 through the Rational Root Theorem and synthetic division. This step-by-step solution includes solving a quadratic equation with complex roots. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation