Questions: Quiz: Practice Questions for Final Test Analytios Ryan Find all roots of h(x). h(x) = -x^3 + 3x^2 - 4x + 2 Write your answer as a list of simplified values separated by commas, if there is more than one value.

Solution Steps

The polynomial \( h(x) = -x^{3} + 3x^{2} - 4x + 2 \) can be factorized as: \[ h(x) = - (x - 1)(x^{2} - 2x + 2) \]

To find the roots of the polynomial, we set \( h(x) = 0 \): \[

• (x - 1)(x^{2} - 2x + 2) = 0 \] This gives us two factors to consider: \( x - 1 = 0 \) and \( x^{2} - 2x + 2 = 0 \).

1. From the first factor \( x - 1 = 0 \), we find: \[ x = 1 \]

2. For the second factor \( x^{2} - 2x + 2 = 0 \), we apply the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} = \frac{2 \pm \sqrt{(-2)^{2} - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 - 8}}{2} = \frac{2 \pm \sqrt{-4}}{2} \] Simplifying this, we get: \[ x = 1 \pm i \]

The roots of the polynomial \( h(x) \) are: \[ x = 1, \quad x = 1 - i, \quad x = 1 + i \]

Final Answer