Given that one of the zeroes of the cubic polynomial ax³ + bx² + cx + d is zero, find the product of the other two zeroes

Video Explanation

Watch the video explanation below:

Given

f(x) = ax³ + bx² + cx + d

One of the zeroes of the polynomial is 0.

To Find

The product of the other two zeroes.

Solution

Let the three zeroes of the polynomial be:

0, α and β

For a cubic polynomial ax³ + bx² + cx + d, the relationships between zeroes and coefficients are:

Sum of zeroes = −b/a

Sum of the product of zeroes taken two at a time = c/a

Product of all three zeroes = −d/a

Step 1: Use the Given Information

Since one zero is 0,

Product of all three zeroes = 0 × α × β = 0

So,

−d/a = 0

⇒ d = 0

Step 2: Find the Product of the Other Two Zeroes

Sum of the product of zeroes taken two at a time is:

αβ + 0·α + 0·β = αβ

But,

αβ = c/a

Final Answer

The product of the other two zeroes is c/a.

Conclusion

Hence, if one of the zeroes of the cubic polynomial ax³ + bx² + cx + d is zero, then the product of the other two zeroes is c/a.