Quadratic Polynomial with Opposite Zeroes

Video Explanation

Question

If one of the zeroes of a quadratic polynomial of the form

\[ x^2 + ax + b \]

is the negative of the other, then it:

(a) has no linear term and constant term is negative
(b) has no linear term and the constant term is positive
(c) can have a linear term but the constant term is negative
(d) can have a linear term but the constant term is positive

Solution

Step 1: Assume the Zeroes

Let the zeroes be \( \alpha \) and \( -\alpha \).

Step 2: Use Relations Between Zeroes and Coefficients

For the polynomial \(x^2 + ax + b\),

Sum of zeroes:

\[ \alpha + (-\alpha) = 0 \]

But sum of zeroes is also equal to:

\[ -a \]

Hence,

\[ -a = 0 \Rightarrow a = 0 \]

So, the polynomial has no linear term.

Step 3: Find the Sign of the Constant Term

Product of zeroes:

\[ \alpha \times (-\alpha) = -\alpha^2 \]

But product of zeroes is also equal to:

\[ b \]

Since \( -\alpha^2 < 0 \), we get:

\[ b < 0 \]

Conclusion

The polynomial:

• has no linear term, and
• its constant term is negative.

Hence, the correct option is:

\[ \boxed{\text{(a)}} \]