what are x-intercepts for the equation y=2x^3-x^2+2x-1

To find the x-intercepts of the equation \( y = 2x^3 - x^2 + 2x - 1 \), you need to set \( y \) to 0 and solve for \( x \):

\[ 2x^3 - x^2 + 2x - 1 = 0 \]

This is a cubic equation, and finding its roots can be done through various methods such as graphing, synthetic division, or using the Rational Root Theorem.

1. Rational Root Theorem: Potential rational roots could be factors of the constant term (-1) divided by factors of the leading coefficient (2). The candidates are: \( \pm 1, \pm \frac{1}{2} \).

2. Testing possible roots:

   • Let's test \( x = 1 \): \[ 2(1)^3 - (1)^2 + 2(1) - 1 = 2 - 1 + 2 - 1 = 2 \quad (\text{not a root}) \]
   • Now test \( x = -1 \): \[ 2(-1)^3 - (-1)^2 + 2(-1) - 1 = -2 - 1 - 2 - 1 = -6 \quad (\text{not a root}) \]
   • Now test \( x = \frac{1}{2} \): \[ 2\left(\frac{1}{2}\right)^3 - \left(\frac{1}{2}\right)^2 + 2\left(\frac{1}{2}\right) - 1 = 2\left(\frac{1}{8}\right) - \frac{1}{4} + 1 - 1 \] \[ = \frac{1}{4} - \frac{1}{4} + 1 - 1 = 0 \quad (\text{is a root}) \]

Since \( x = \frac{1}{2} \) is a root, we can use synthetic division to divide the polynomial \( 2x^3 - x^2 + 2x - 1 \) by \( x - \frac{1}{2} \):

Synthetic Division:

0.5 |  2  -1   2  -1
    |      1   0.0  1.0
    ------------------
      2  0.0  2.0  0

The result is \( 2x^2 + 0x + 2 \).

Factoring the Remaining Quadratic:

We now solve the quadratic \( 2x^2 + 2 = 0 \):

\[ 2x^2 = -2 \quad \rightarrow \quad x^2 = -1 \quad \rightarrow \quad x = i, -i \]

Conclusion:

The x-intercepts of the equation \( y = 2x^3 - x^2 + 2x - 1 \) is \( \left(\frac{1}{2}, 0\right) \) while \( x = i \) and \( x = -i \) are complex roots. Therefore, the only x-intercept in the real number set is:

\[ \text{x-intercept: } \left(\frac{1}{2}, 0\right) \]