The zeros of the cubic polynomial `2x^(3) -x^(2)-2x + 1` are a, b and c respectively. Find ab + bc + ca=?

Options

A. 2
B. 0
C. `+1`
D. `-1`

Answer

To find the value of \( ab + bc + ca \) for the cubic polynomial \( 2x^3 - x^2 - 2x + 1 \), we can use Vieta's formulas. Vieta's relations give us a way to relate the coefficients of a polynomial to sums and products of its roots.

### Step-by-Step Solution:

1. **Identify the coefficients of the polynomial**:
The given polynomial is \( 2x^3 - x^2 - 2x + 1 \). Here, we can identify the coefficients:
- \( a = 2 \) (coefficient of \( x^3 \))
- \( b = -1 \) (coefficient of \( x^2 \))
- \( c = -2 \) (coefficient of \( x \))
- \( d = 1 \) (constant term)

2. **Apply Vieta's formulas**:
According to Vieta's formulas for a cubic polynomial \( ax^3 + bx^2 + cx + d \), the sum of the products of the roots taken two at a time (which is what we need to find, \( ab + ac + bc \)) is given by:
\[ ab + ac + bc = \frac{c}{a} \]

3. **Substitute the coefficients into the formula**:
Here, \( c = -2 \) and \( a = 2 \). Therefore:
\[ ab + ac + bc = \frac{-2}{2} \]

4. **Calculate the result**:
Simplifying the right-hand side gives:
\[ ab + ac + bc = -1 \]

5. **Conclusion**:
Thus, the value of \( ab + ac + bc \) is \( -1 \).

### Final Answer:
\[ ab + ac + bc = -1 \]