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Solve the polynomial equation by factoring and check the solutions graphically. xSuperscript 4minus8xcubedplus15xsquaredequals0...Question
Solve the polynomial equation by factoring and check the solutions graphically.
xSuperscript 4minus8xcubedplus12xsquaredequals0
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Part 1
The solutions are
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xSuperscript 4minus8xcubedplus12xsquaredequals0
Question content area bottom
Part 1
The solutions are
enter your response here.
(Simplify your answer. Use a comma to separate answers as needed.)
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Answered by
GPT-4o mini
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 \( x = 6 \).
\[
x^2(x^2 - 8x + 12) = 0
\]
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 \( x = 6 \).
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