Finding the Third Zero of a Cubic Polynomial

Video Explanation

Question

If two of the zeroes of the cubic polynomial

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

are each equal to zero, find the third zero.

Solution

Step 1: Use the Product of Zeroes Formula

For a cubic polynomial

\[ ax^3 + bx^2 + cx + d, \]

the product of its zeroes is:

\[ -\frac{d}{a} \]

Step 2: Substitute the Given Zeroes

Let the three zeroes be:

\[ 0,\; 0,\; \gamma \]

Then,

\[ 0 \times 0 \times \gamma = 0 \]

Step 3: Equate with the Product of Zeroes

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

This is possible only if:

\[ d = 0 \]

Hence, the remaining factor of the polynomial is \(x\).

Step 4: Find the Third Zero

The third zero is obtained from:

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

So, the third zero is:

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

Conclusion

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

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