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:
-
\( x^2 = 0 \)
- \( x = 0 \)
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\( x - 3 = 0 \)
- \( x = 3 \)
-
\( x - 5 = 0 \)
- \( x = 5 \)
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.
- The graph should touch the x-axis at \( x = 0 \) which will be a double root (indicating that it touches and turns).
- 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 \( x = 0, 3, 5 \).
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