If one of the zeroes of the cubic polynomial x³ + ax² + bx + c is −1, find the product of the other two zeroes

Video Explanation

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Given

Cubic polynomial: f(x) = x³ + ax² + bx + c

One zero of the polynomial is −1.

To Find

The product of the other two zeroes.

Solution

Since −1 is a zero of the polynomial, by the Factor Theorem:

f(−1) = 0

Step 1: Substitute x = −1

(−1)³ + a(−1)² + b(−1) + c = 0

−1 + a − b + c = 0

∴ c = b − a + 1

Step 2: Use Relationship Between Zeroes and Coefficients

Let the three zeroes be −1, α and β.

For a cubic polynomial:

Product of all zeroes = −c

∴ (−1)(αβ) = −c

⇒ αβ = c

Step 3: Substitute the Value of c

αβ = b − a + 1

Final Answer

The product of the other two zeroes is:

b − a + 1

Correct Option

(a) b − a + 1

Conclusion

Hence, if one of the zeroes of the cubic polynomial x³ + ax² + bx + c is −1, then the product of the other two zeroes is b − a + 1.