Nature of Zeroes of a Quadratic Polynomial

Video Explanation

Question

The zeroes of the quadratic polynomial

\[ f(x) = x^2 + 99x + 127 \]

are:

(a) both positive
(b) both negative
(c) both equal
(d) one positive and one negative

Solution

Step 1: Use Relations Between Zeroes and Coefficients

For a quadratic polynomial \[ ax^2 + bx + c, \]

\[ \text{Sum of zeroes} = -\frac{b}{a}, \quad \text{Product of zeroes} = \frac{c}{a} \]

Step 2: Apply to the Given Polynomial

Here,

\[ a = 1,\quad b = 99,\quad c = 127 \]

Sum of zeroes:

\[ -\frac{99}{1} = -99 \; (< 0) \]

Product of zeroes:

\[ \frac{127}{1} = 127 \; (> 0) \]

Step 3: Decide the Nature of Zeroes

• Sum of zeroes is negative
• Product of zeroes is positive

Therefore, both zeroes are negative.

Conclusion

The correct answer is:

\[ \boxed{\text{both negative}} \]

Hence, the correct option is (b).