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 .

Answer 1

Let the quadratic polynomial be $f\left(x\right)=x^2+ax+b$.
Now, the zeroes are $\alpha \mathrm{and}-\alpha$.
So, the sum of the zeroes is zero.
$$\begin{gathered} \therefore \alpha +\left(-\alpha \right)=\frac{-a}{1}=-a \\ \Rightarrow a=0 \end{gathered}$$
So, the polynomial becomes $f\left(x\right)=x^2+b$, which is not linear
Also, the product of the zeros,
$$\begin{gathered} \alpha \beta =\frac{b}{1}=b \\ \Rightarrow \alpha \left(-\alpha \right)=b \\ \Rightarrow -{\alpha }^2=b \end{gathered}$$
Thus, the constant term is negative.

Hence, the correct answer is option A.