Question

determine the nature of the roots of the polynomial equation:

2x^3 - 5x^2 + 5x - 2 = 0

Answer 1

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 equations. However, it can be simpler to analyze the function's behavior through its derivative and then check any potential rational roots first.

1. **Finding the derivative**:
\[
f(x) = 2x^3 - 5x^2 + 5x - 2
\]
\[
f'(x) = 6x^2 - 10x + 5
\]

2. **Finding critical points** by setting the derivative to zero:
\[
6x^2 - 10x + 5 = 0
\]
To solve, we find the discriminant:
\[
D = (-10)^2 - 4 \cdot 6 \cdot 5 = 100 - 120 = -20
\]
Since the discriminant is negative, there are no real critical points, which indicates that \(f(x)\) is strictly increasing.

3. **Checking values at specific points**:
- \(f(0) = -2\) (the function is negative)
- \(f(1) = 2(1)^3 - 5(1)^2 + 5(1) - 2 = 2 - 5 + 5 - 2 = 0\) (the function has a root at \(x = 1\))
- \(f(2) = 2(2)^3 - 5(2)^2 + 5(2) - 2 = 16 - 20 + 10 - 2 = 4\) (the function is positive)

4. **Roots behavior**:
Since \(f(0) < 0\) and \(f(1) = 0\) (a root), and \(f(2) > 0\), and considering that the polynomial is strictly increasing, there is exactly one real root at \(x = 1\).

5. **Finding the nature of other roots**:
The total number of roots in a cubic polynomial is 3 (counting multiplicities). Since we have one real root, the other roots must be a pair of complex conjugates (since coefficients are real).

Therefore, the nature of the roots of the polynomial \(2x^3 - 5x^2 + 5x - 2 = 0\) consists of one real root and two complex conjugate roots.