Solve the polynomial equation by using the root​ method, and check the solutions graphically.

Question

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: \(5, -5\)

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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.