Find all the real zeros of the polynomial. Separate answers by commas. Enter exact answers using integers, fractions and radicals, not decimal approximations.
f
(
x
)
=
   
x
5
−
6
x
4
−
36
x
+
216
   
Zeros:  
x
  =

The given polynomial is:

\[
f(x) = x^5 - 6x^4 - 36x + 216
\]

To find the real zeros, we need to factor the polynomial or use techniques such as synthetic division, the Rational Root Theorem, or trial and error to check possible roots. Let's go through the process:

### Step 1: Try Rational Root Theorem
The Rational Root Theorem suggests that any possible rational root should be a factor of the constant term (216) divided by a factor of the leading coefficient (1, since the coefficient of \(x^5\) is 1).

Factors of 216 include: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 9, \pm 12, \pm 18, \pm 24, \pm 27, \pm 36, \pm 54, \pm 72, \pm 108, \pm 216 \).

We will check for possible roots.

### Step 2: Test possible roots
Testing \(x = 6\):

\[
f(6) = 6^5 - 6(6^4) - 36(6) + 216
\]
\[
f(6) = 7776 - 6(1296) - 216 + 216 = 7776 - 7776 = 0
\]

Thus, \(x = 6\) is a root.

### Step 3: Perform synthetic division
Now that we know \(x = 6\) is a root, we can divide \(f(x)\) by \(x - 6\) using synthetic division.

Dividing \(x^5 - 6x^4 - 36x + 216\) by \(x - 6\) yields:

\[
x^5 - 6x^4 - 36x + 216 = (x - 6)(x^4 - 36)
\]

### Step 4: Factor \(x^4 - 36\)
The expression \(x^4 - 36\) is a difference of squares:

\[
x^4 - 36 = (x^2 - 6)(x^2 + 6)
\]

### Step 5: Solve each factor
Now we solve for the zeros of each factor:

1. \(x - 6 = 0 \Rightarrow x = 6\)
2. \(x^2 - 6 = 0 \Rightarrow x = \pm \sqrt{6}\)
3. \(x^2 + 6 = 0 \Rightarrow x = \pm \sqrt{-6} \Rightarrow \text{No real solutions} \) (since \(\sqrt{-6}\) is imaginary)

### Step 6: Real zeros
The real zeros of the polynomial are:

\[
x = 6, \, x = \sqrt{6}, \, x = -\sqrt{6}
\]

### Final Answer:
\[
x = 6, \sqrt{6}, -\sqrt{6}
\]

Would you like more details on any step? Here are some questions to deepen understanding:

1. How does the Rational Root Theorem work for polynomials?
2. What are the steps involved in synthetic division?
3. Why is \(x^4 - 36\) a difference of squares?
4. How can we solve higher-degree polynomials?
5. What is the significance of imaginary versus real roots?

**Tip:** Always test smaller integer roots first using the Rational Root Theorem to simplify polynomials.

Find all the real zeros of the polynomial. Separate answers by commas. Enter exact answers using integers, fractions and radicals, not decimal approximations.
f(x) = x^5 - 6x^4 - 36x + 216
Zeros: x = 

Learn how to find the real zeros of the polynomial x^5 - 6x^4 - 36x + 216 using methods like the Rational Root Theorem, synthetic division, and factorization. This problem covers advanced algebraic concepts and is suitable for students in Grades 10-12.

Math Problem Statement

Find all the real zeros of the polynomial. Separate answers by commas. Enter exact answers using integers, fractions and radicals, not decimal approximations. f ( x )

Zeros:
x

Solution

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