Finding the Third Zero of a Cubic Polynomial

Video Explanation

Question

Given that two of the zeroes of the cubic polynomial

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

are \(0\), find the third zero.

Solution

Step 1: Assume the Zeroes

Let the zeroes of the polynomial be:

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

Step 2: Use the Product of Zeroes Formula

For a cubic polynomial,

\[ \text{Product of zeroes} = -\frac{d}{a} \]

So,

\[ 0 \times 0 \times \alpha = -\frac{d}{a} \]

\[ 0 = -\frac{d}{a} \Rightarrow d = 0 \]

Step 3: Find the Third Zero

With \(d = 0\), the polynomial becomes:

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

Hence, the third zero is obtained from:

\[ ax^2 + bx + c = 0 \]

So, the third zero is:

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

Conclusion

If two zeroes of the cubic polynomial are zero, then the third zero is:

\[ \boxed{-\dfrac{c}{a}} \]