To determine the nature of the roots of the polynomial equation \( 2x^3 - 5x^2 + 5x - 2 = 0 \), we can use the discriminant for cubic polynomials, but a more straightforward approach might be to use the Rational Root Theorem and the first derivative to analyze the function.
Step 1: Find Critical Points
We first find the derivative of the polynomial:
\[ f(x) = 2x^3 - 5x^2 + 5x - 2 \] \[ f'(x) = 6x^2 - 10x + 5 \]
Step 2: Determine the number of critical points
To find the critical points, we set \( f'(x) = 0 \):
\[ 6x^2 - 10x + 5 = 0 \]
We can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 6 \cdot 5}}{2 \cdot 6} \] \[ x = \frac{10 \pm \sqrt{100 - 120}}{12} \] \[ x = \frac{10 \pm \sqrt{-20}}{12} \] \[ x = \frac{10 \pm 2i\sqrt{5}}{12} \] \[ x = \frac{5 \pm i\sqrt{5}}{6} \]
Since the critical points are complex, it indicates that \( f(x) \) does not change direction and is either strictly increasing or strictly decreasing.
Step 3: Analyzing Crossings of the X-Axis
Now we evaluate the polynomial at a couple of points to determine the sign of the function for large positive and negative values:
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For \( x = -1 \): \[ f(-1) = 2(-1)^3 - 5(-1)^2 + 5(-1) - 2 = -2 - 5 - 5 - 2 = -14 < 0 \]
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For \( x = 0 \): \[ f(0) = -2 < 0 \]
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For \( x = 1 \): \[ f(1) = 2(1)^3 - 5(1)^2 + 5(1) - 2 = 2 - 5 + 5 - 2 = 0 \]
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For \( x = 2 \): \[ f(2) = 2(2)^3 - 5(2)^2 + 5(2) - 2 = 16 - 20 + 10 - 2 = 4 > 0 \]
Conclusion
We have found:
- \( f(1) = 0 \) meaning \( x = 1 \) is a root.
- \( f(-1) < 0 \) and \( f(2) > 0 \) indicate there is a sign change, confirming there is another root between (0, 2).
The polynomial is strictly increasing (as indicated by the complex critical points), so it can only cross the x-axis at most 3 times. Since there is one real root confirmed at \( x=1 \) and at least one other real root between 0 and 2, hence there are likely three real roots ranging in nature.
Final Answer: Three real roots.
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