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 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.

Answer 1

Let $p\left(x\right)=x^2+ax+b$
Now, product of zeroes = $\frac{\text{constant}}{\text{coefficient of}{x}^2}$
Let $\alpha \text{and}\beta$ be the zeroes of $p\left(x\right)$
$\Rightarrow$ product of zeroes $\left(\alpha \beta \right)=\frac{b}{1}$
$\Rightarrow \alpha \beta =b$
Given that, one of the zeroes of a quadratic polynomial $p\left(x\right)$ is negative of the other
$\Rightarrow \alpha \beta <0$
So, $b<0$
Hence, $b$ should be negative
Put $a=0$, then, $p\left(x\right)=x^2+b=0$
$\Rightarrow x^2=-b$
$\Rightarrow a=0$
$\therefore$ $f\left(x\right)=x^2+b$ , which cannot be linear
And product of zeroes = $\alpha (-\alpha )=b$
$\Rightarrow$ $-a^2=b$
Which is possible when $b<0$
So, it has no linear term and the constant term is negative.

Hence, Option A is correct.