If `alpha, beta and lambda` be the roots of the cubic equation `ax^3 + bx^2 + cx + d =0` then find the product of the roots

Answer

To find the product of the roots \( \alpha, \beta, \lambda \) of the cubic equation \( ax^3 + bx^2 + cx + d = 0 \), we can use Viète's formulas, which relate the coefficients of the polynomial to sums and products of its roots.

### Step-by-Step Solution:

1. **Identify the given cubic equation**:
The cubic equation is given as:
\[ ax^3 + bx^2 + cx + d = 0 \]
where \( a, b, c, d \) are constants, and \( \alpha, \beta, \lambda \) are the roots.

2. **Use Viète's formulas**:
According to Viète's formulas for a cubic equation \( ax^3 + bx^2 + cx + d = 0 \):
- The sum of the roots \( \alpha + \beta + \lambda = -\frac{b}{a} \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\lambda + \alpha\lambda = \frac{c}{a} \)
- The product of the roots \( \alpha\beta\lambda = -\frac{d}{a} \)

3. **Find the product of the roots**:
From Viète's formulas, we can directly find the product of the roots:
\[ \alpha\beta\lambda = -\frac{d}{a} \]

4. **Conclusion**:
Therefore, the product of the roots \( \alpha, \beta, \lambda \) of the cubic equation \( ax^3 + bx^2 + cx + d = 0 \) is:
\[ \alpha\beta\lambda = -\frac{d}{a} \]