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

Question

xSuperscript 4minus6xcubedplus8xsquaredequals0
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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 \( x^4 - 6x^3 + 8x^2 = 0 \) by factoring, we can start by factoring out the common term.

1. Factor out the common term \( x^2 \): \[ x^2(x^2 - 6x + 8) = 0 \]

2. Now we have two factors to set to zero:

   • \( x^2 = 0 \)
   • \( x^2 - 6x + 8 = 0 \)

3. Solve \( x^2 = 0 \):

   • The solution is \( x = 0 \).

4. Now solve the quadratic equation \( x^2 - 6x + 8 = 0 \). We can factor this quadratic: \[ x^2 - 6x + 8 = (x - 2)(x - 4) = 0 \]

5. Set each factor to zero:

   • \( x - 2 = 0 \) gives \( x = 2 \).
   • \( x - 4 = 0 \) gives \( x = 4 \).

6. Combine all the solutions: The complete set of solutions is: \[ x = 0, 2, 4 \]

Final answer: The solutions are \( 0, 2, 4 \).

Graphical Check

To check these solutions graphically, you can plot the function \( y = x^4 - 6x^3 + 8x^2 \) and see where it intersects the x-axis. These intersection points correspond to the solutions of the polynomial equation. You'll find that the graph touches the x-axis at \( x = 0 \), \( x = 2 \), and \( x = 4 \), confirming that these are indeed the solutions.