To find the full solution set for the polynomial equation \(x^3 + 5x^2 + 10x + 50 = 0\) given that one of the solutions is \(x = -5\), we can perform synthetic division to factor the polynomial.
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Perform synthetic division with \(x + 5\):
Set \(x = -5\) for synthetic division:
-5 | 1 5 10 50 | -5 0 -50 --------------------- 1 0 10 0The result is \(x^2 + 10\) with a remainder of 0.
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Factor the equation:
Now, we have: \[ x^3 + 5x^2 + 10x + 50 = (x + 5)(x^2 + 10) \]
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Solve \(x^2 + 10 = 0\):
\[ x^2 = -10 \]
The solutions are: \[ x = \sqrt{-10} = \pm i\sqrt{10} \]
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Full solution set:
The solution set includes:
- One real solution: \(x = -5\)
- Two non-real solutions: \(x = i\sqrt{10}\) and \(x = -i\sqrt{10}\)
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Choose the correct option:
The correct option that describes this solution set is:
Option #1: It has one real solution and two non-real solutions.
1 answer