If one of the zeros of a quadratic polynomial of the form x² + ax + b is the negative of the other, then find the correct statement

Video Explanation

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Given

Quadratic polynomial: f(x) = x² + ax + b

One zero is the negative of the other.

To Find

The correct statement about the polynomial.

Solution

Let the zeroes of the polynomial be:

α and −α

Step 1: Use the Sum of Zeroes

Sum of zeroes = α + (−α) = 0

But for the polynomial x² + ax + b:

Sum of zeroes = −a

So,

−a = 0

⇒ a = 0

Hence, the polynomial has no linear term.

Step 2: Use the Product of Zeroes

Product of zeroes = α(−α) = −α²

But for the polynomial:

Product of zeroes = b

So,

b = −α²

Since α² > 0, we get:

b < 0

Final Answer

The polynomial has no linear term and the constant term is negative.

Correct Option

(a) has no linear term and constant term is negative

Conclusion

Hence, if one zero of the quadratic polynomial x² + ax + b is the negative of the other, then the polynomial has no linear term and its constant term is negative.