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 the 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
Step-by-Step Solution
Key Concept: If zeroes are $\alpha$ and $-\alpha$, sum of zeroes $= \alpha + (-\alpha) = 0 \Rightarrow a = 0$. Product $= -\alpha^2 < 0 \Rightarrow b < 0$.
Let zeroes be $\alpha$ and $-\alpha$. Sum $= 0 \Rightarrow -a = 0 \Rightarrow a = 0$ (no linear term). [0.5 Mark]
Product $= \alpha(-\alpha) = -\alpha^2 = b$. Since $\alpha^2 > 0$, $b < 0$ (constant term is negative). [0.5 Mark]
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🎯 Official CBSE Marking Scheme:
Deducing sum = 0 and linear term coefficient $a = 0$: 0.5 Mark
Deducing product $< 0$ and constant term $b < 0$: 0.5 Mark
Correct Answer: Has no linear term and the constant term is negative