Question

etermine the nature of the roots of the polynomial equation 2 x cubed minus 5 x squared plus 5 x minus 2 equals 0.
(1 point)
Responses

three real roots
three real roots

two real roots and one complex root
two real roots and one complex root

one real root and two complex roots
one real root and two complex roots

three complex roots

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 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:

- For \( x = -1 \):
\[
f(-1) = 2(-1)^3 - 5(-1)^2 + 5(-1) - 2 = -2 - 5 - 5 - 2 = -14 < 0
\]

- For \( x = 0 \):
\[
f(0) = -2 < 0
\]

- For \( x = 1 \):
\[
f(1) = 2(1)^3 - 5(1)^2 + 5(1) - 2 = 2 - 5 + 5 - 2 = 0
\]

- 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.