Question

Solve the polynomial equation by using the root​ method, and check the solutions graphically.
one third x Superscript 4 Baseline minus 27 equals 0
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Part 1
xequals
  
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Answer 1

To solve the polynomial equation

\[
\frac{1}{3} x^4 - 27 = 0,
\]

we first isolate \(x^4\):

\[
\frac{1}{3} x^4 = 27.
\]

Next, we multiply both sides by 3 to eliminate the fraction:

\[
x^4 = 81.
\]

Now, we take the fourth root of both sides:

\[
x = \pm 81^{1/4}.
\]

Calculating \(81^{1/4}\):

\[
81^{1/4} = (3^4)^{1/4} = 3.
\]

Thus, the solutions are:

\[
x = 3 \quad \text{and} \quad x = -3.
\]

So, the complete solution set is:

\[
x = 3, -3.
\]

Now, let's write that in the requested format:

\[
x = 3, -3.
\]

### Graphical Check

To check the solutions graphically, you can plot the function

\[
f(x) = \frac{1}{3} x^4 - 27
\]

and observe where the graph intersects the x-axis (where \(f(x) = 0\)). You should see that there are roots at \(x = 3\) and \(x = -3\).