If one of the zeros of the polynomial $x^3 + ax^2 + bx + c$ is −1, then the product of other two zeros is equal to :

Question

Accepted Answer

To solve this problem, we need to determine the product of the other two zeros of the polynomial $x^3 + ax^2 + bx + c$ given that one of the zeros is $-1$.

### Explanation:

1. **Given Zero:**
   - One of the zeros of the polynomial is $-1$.

2. **Vieta's Formulas:**
   - For a cubic polynomial $x^3 + ax^2 + bx + c$, Vieta's formulas tell us:
     - The sum of the zeros, $\alpha + \beta + \gamma = -a$.
     - The sum of the products of the zeros taken two at a time, $\alpha\beta + \beta\gamma + \gamma\alpha = b$.
     - The product of the zeros, $\alpha\beta\gamma = -c$.

3. **Substituting the Known Zero:**
   - Let the zeros be $-1$, $\beta$, and $\gamma$.
   - According to Vieta's formulas, $-1 + \beta + \gamma = -a$.
   - This simplifies to $\beta + \gamma = -a + 1$.

4. **Product of the Other Two Zeros:**
   - We need to find the product $\beta\gamma$.
   - From Vieta's formulas, $\alpha\beta + \beta\gamma + \gamma\alpha = b$.
   - Substituting $\alpha = -1$, we have:
     $$
     (-1)\beta + \beta\gamma + (-1)\gamma = b
     $$
     $$
     -\beta - \gamma + \beta\gamma = b
     $$
   - We already know $\beta + \gamma = -a + 1$, so:
     $$
     -(-a + 1) + \beta\gamma = b
     $$
     $$
     a - 1 + \beta\gamma = b
     $$
   - Solving for $\beta\gamma$, we get:
     $$
     \beta\gamma = b - a + 1
     $$

### Conclusion:

- The product of the other two zeros $\beta\gamma$ is indeed $b - a + 1$, which corresponds to **Option 1**.

### Why Other Options are Incorrect:

- **Option 2 (b - a - 1):** This option would imply a different relationship between the coefficients and the zeros, which does not align with the derived equation $\beta\gamma = b - a + 1$.

- **Option 3 (a - b + 1):** This option incorrectly rearranges the terms and does not satisfy the equation derived from Vieta's formulas.

- **Option 4 (a - b - 1):** Similar to Option 3, this option does not match the correct expression for $\beta\gamma$ as derived from the polynomial's properties.

Thus, the correct answer is **Option 1: b - a + 1**.