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

Question

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 \(x = -3\).