Question

Solve the polynomial equation by using the root​ method, and check the solutions graphically.
one fifth x Superscript 4 Baseline minus 125 equals 0
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Part 1
xequals
  
enter your response here ​(Type an integer or a decimal. Use a comma to separate answers as​ needed.)

Answer 1

To solve the polynomial equation \(\frac{1}{5} x^4 - 125 = 0\), we will first isolate the variable.

1. **Multiply both sides by 5 to eliminate the fraction:**

\[
x^4 - 625 = 0
\]

2. **Add 625 to both sides:**

\[
x^4 = 625
\]

3. **Take the fourth root of both sides:**

\[
x = \pm \sqrt[4]{625}
\]

Since \(625 = 25^2\), we can further simplify:

\[
x = \pm \sqrt{25} = \pm 5
\]

So the solutions are:

\[
x = 5 \quad \text{and} \quad x = -5
\]

Thus, the final answer is:

\[
x = 5, -5
\]

**Part 1 Response:**
\(5, -5\)

---

### Graphical Check

To check the solutions graphically, you can plot the function \(y = \frac{1}{5} x^4 - 125\) and observe where it intersects the x-axis. The points where the graph intersects the x-axis will confirm the roots \(x = 5\) and \(x = -5\).

Using graphing software or a graphing calculator, you can input:

\[
f(x) = \frac{1}{5}x^4 - 125
\]

When you check the graph, you should see that the curve intersects the x-axis at \(x = 5\) and \(x = -5\), confirming our solutions.