https://r2.erweima.ai/mathbot/image/17189901551101wtly4uz.png

To find all zeros of the polynomial function \( f(x) = x^4 - 6x^3 + 7x^2 + 6x - 8 \), we will follow these steps:

1. **Check for Rational Roots**: Use the Rational Root Theorem to identify possible rational roots.
2. **Test Possible Roots**: Substitute the possible rational roots into the polynomial to see if they are actual roots.
3. **Factor the Polynomial**: Once a root is found, factor the polynomial using synthetic division or polynomial division.
4. **Solve the Reduced Polynomial**: Repeat the process on the reduced polynomial until all roots are found.

### Step 1: Rational Root Theorem

The Rational Root Theorem states that any rational root of the polynomial equation \( f(x) = 0 \) is of the form \( \frac{p}{q} \), where:
- \( p \) is a factor of the constant term \( a_0 = -8 \).
- \( q \) is a factor of the leading coefficient \( a_n = 1 \).

The factors of \(-8\) are \( \pm 1, \pm 2, \pm 4, \pm 8 \).
The factors of \(1\) are \( \pm 1 \).

Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 4, \pm 8 \).

### Step 2: Test Possible Roots

We test the possible roots by substituting them into the polynomial \( f(x) \).

- \( f(1) = 1^4 - 6(1)^3 + 7(1)^2 + 6(1) - 8 = 1 - 6 + 7 + 6 - 8 = 0 \)

So, \( x = 1 \) is a root.

### Step 3: Factor the Polynomial

Since \( x = 1 \) is a root, we can factor \( (x - 1) \) out of the polynomial.

We perform synthetic division using \( x = 1 \):

\[
\begin{array}{r|rrrrr}
1 & 1 & -6 & 7 & 6 & -8 \\
\hline
 & 1 & -5 & 2 & 8 & 0 \\
\end{array}
\]

The quotient is \( x^3 - 5x^2 + 2x + 8 \).

### Step 4: Solve the Reduced Polynomial

Now, we solve the reduced polynomial \( x^3 - 5x^2 + 2x + 8 \).

Using the Rational Root Theorem again, we test possible roots \( \pm 1, \pm 2, \pm 4, \pm 8 \):

- \( f(2) = 2^3 - 5(2)^2 + 2(2) + 8 = 8 - 20 + 4 + 8 = 0 \)

So, \( x = 2 \) is a root.

We perform synthetic division again using \( x = 2 \):

\[
\begin{array}{r|rrrr}
2 & 1 & -5 & 2 & 8 \\
\hline
 & 1 & -3 & -4 & 0 \\
\end{array}
\]

The quotient is \( x^2 - 3x - 4 \).

We solve \( x^2 - 3x - 4 = 0 \) by factoring:

\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\]

Thus, the roots are \( x = 4 \) and \( x = -1 \).

### Summary of Roots

The roots of the polynomial \( f(x) = x^4 - 6x^3 + 7x^2 + 6x - 8 \) are:
\[ x = 1, 2, 4, -1 \]

Would you like further details or have any questions?

Here are 5 additional questions you might consider:

1. How do you use the Rational Root Theorem for other polynomial equations?
2. Can you explain synthetic division in more detail?
3. How do you factor quadratic equations efficiently?
4. What methods are available for finding complex roots?
5. Can polynomial equations always be factored into linear factors?

**Tip:** When factoring polynomials, always start by looking for the greatest common factor (GCF) to simplify the process.

Learn how to find all zeros of the polynomial function x^4 - 6x^3 + 7x^2 + 6x - 8 using the Rational Root Theorem and synthetic division. This problem covers polynomial functions, root finding strategies, and synthetic division techniques. Suitable 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