Product of Zeroes of a Cubic Polynomial

Video Explanation

Question

Given that one of the zeroes of the cubic polynomial

\[ f(x) = ax^3 + bx^2 + cx + d \]

is zero, find the product of the other two zeroes.

Solution

Step 1: Write Relations Between Zeroes and Coefficients

Let the zeroes of the polynomial be

\[ 0,\; \alpha,\; \beta \]

For a cubic polynomial,

\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]

Step 2: Substitute One Zero as 0

\[ (0 \cdot \alpha) + (0 \cdot \beta) + (\alpha \beta) = \alpha \beta \]

So,

\[ \alpha \beta = \frac{c}{a} \]

Conclusion

The product of the other two zeroes is:

\[ \boxed{\frac{c}{a}} \]