Solve the polynomial equation by factoring and check the solutions graphically.

Question

xSuperscript 4minus8xcubedplus12xsquaredequals0
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Part 1
The solutions are
  
enter your response here.
​(Simplify your answer. Use a comma to separate answers as​ needed.)

Answer 1

To solve the polynomial equation \( x^4 - 8x^3 + 12x^2 = 0 \) by factoring, we first factor out the greatest common factor (GCF), which is \( x^2 \):

\[ x^2(x^2 - 8x + 12) = 0 \]

Next, we set each factor equal to zero:

1. For \( x^2 = 0 \): \[ x = 0 \]

2. For the quadratic part \( x^2 - 8x + 12 = 0 \), we can factor it further: \[ x^2 - 8x + 12 = (x - 6)(x - 2) \]

Now we set each factor to zero:

• From \( x - 6 = 0 \): \[ x = 6 \]

• From \( x - 2 = 0 \): \[ x = 2 \]

So, the complete set of solutions to the equation \( x^4 - 8x^3 + 12x^2 = 0 \) is:

\[ x = 0, 2, 6 \]

Thus, the solutions are:

\[ \boxed{0, 2, 6} \]

Checking Graphically

To check the solutions graphically, you can plot the function \( f(x) = x^4 - 8x^3 + 12x^2 \) and look for the points where the graph intersects the x-axis. The x-intercepts represent the solutions to the equation. The graph should confirm that there are roots at \( x = 0 \), \( x = 2 \), and \( x = 6 \).