Question
Solve the polynomial equation by factoring and check the solutions graphically.
xSuperscript 4minus8xcubedplus15xsquaredequals0
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Part 1
The solutions are
enter your response here.
(Simplify your answer. Use a comma to separate answers as needed
xSuperscript 4minus8xcubedplus15xsquaredequals0
Question content area bottom
Part 1
The solutions are
enter your response here.
(Simplify your answer. Use a comma to separate answers as needed
Answers
Answered by
GPT-4o mini
To solve the polynomial equation \( x^4 - 8x^3 + 15x^2 = 0 \) by factoring, we can first factor out the greatest common factor, which is \( x^2 \):
\[
x^2(x^2 - 8x + 15) = 0
\]
Next, we can focus on the quadratic \( x^2 - 8x + 15 \) and factor it further. We need two numbers that multiply to \( 15 \) and add to \( -8 \). The numbers \( -3 \) and \( -5 \) satisfy these conditions:
\[
x^2 - 8x + 15 = (x - 3)(x - 5)
\]
Now, we can rewrite the factored form of the equation:
\[
x^2(x - 3)(x - 5) = 0
\]
Setting each factor equal to zero gives us the possible solutions:
1. \( x^2 = 0 \)
- \( x = 0 \)
2. \( x - 3 = 0 \)
- \( x = 3 \)
3. \( x - 5 = 0 \)
- \( x = 5 \)
Thus, the solutions to the polynomial equation are:
\[
x = 0, 3, 5
\]
In summary, the solutions are:
\[
\boxed{0, 3, 5}
\]
### Checking Graphically
To check the solutions graphically, you can plot the original polynomial equation \( y = x^4 - 8x^3 + 15x^2 \). The x-values where the graph intersects the x-axis (y=0) correspond to the roots of the polynomial.
1. The graph should touch the x-axis at \( x = 0 \) which will be a double root (indicating that it touches and turns).
2. The graph will intersect the x-axis at \( x = 3 \) and \( x = 5 \).
If you plot the graph using graphing software or a graphing calculator, the points where the curve crosses the x-axis will confirm the solutions \( x = 0, 3, 5 \).
\[
x^2(x^2 - 8x + 15) = 0
\]
Next, we can focus on the quadratic \( x^2 - 8x + 15 \) and factor it further. We need two numbers that multiply to \( 15 \) and add to \( -8 \). The numbers \( -3 \) and \( -5 \) satisfy these conditions:
\[
x^2 - 8x + 15 = (x - 3)(x - 5)
\]
Now, we can rewrite the factored form of the equation:
\[
x^2(x - 3)(x - 5) = 0
\]
Setting each factor equal to zero gives us the possible solutions:
1. \( x^2 = 0 \)
- \( x = 0 \)
2. \( x - 3 = 0 \)
- \( x = 3 \)
3. \( x - 5 = 0 \)
- \( x = 5 \)
Thus, the solutions to the polynomial equation are:
\[
x = 0, 3, 5
\]
In summary, the solutions are:
\[
\boxed{0, 3, 5}
\]
### Checking Graphically
To check the solutions graphically, you can plot the original polynomial equation \( y = x^4 - 8x^3 + 15x^2 \). The x-values where the graph intersects the x-axis (y=0) correspond to the roots of the polynomial.
1. The graph should touch the x-axis at \( x = 0 \) which will be a double root (indicating that it touches and turns).
2. The graph will intersect the x-axis at \( x = 3 \) and \( x = 5 \).
If you plot the graph using graphing software or a graphing calculator, the points where the curve crosses the x-axis will confirm the solutions \( x = 0, 3, 5 \).