Given that two of the zeroes of the cubic polynomial ax³ + bx² + cx + d are 0, find the third zero

Video Explanation

Watch the video explanation below:

Given

f(x) = ax³ + bx² + cx + d

Two zeroes of the polynomial are:

0 and 0

To Find

The third zero of the polynomial.

Solution

Let the three zeroes of the polynomial be:

0, 0 and α

For a cubic polynomial ax³ + bx² + cx + d, the product of the zeroes is:

−d / a

Step 1: Use the Product of Zeroes

Product of zeroes = 0 × 0 × α = 0

According to the formula:

−d / a = 0

∴ d = 0

Step 2: Factor the Polynomial

Since 0 is a zero of multiplicity 2,

x² is a factor of the polynomial.

So the polynomial can be written as:

f(x) = x²(ax + b)

Step 3: Find the Third Zero

ax + b = 0

∴ x = −b / a

Final Answer

The third zero of the polynomial is −b/a.

Conclusion

Hence, if two of the zeroes of the cubic polynomial ax³ + bx² + cx + d are 0, then the third zero is −b/a.