Question

If one of the zeros 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

The correct option is A has no linear term and the constant term is negative

We know that equation $a{x}^2+bx+c=0$

Then sum of roots $=\frac{-b}{a}$ and product of roots$=\frac{c}{a}$

Given quadratic polynomial is,
$p\left(x\right)=x^2+ax+b$
Let one of the zero of the polynomial is $\alpha$ then the other zero will be $-\alpha$.
Sum of zeros=$\alpha +(-\alpha )=\frac{-a}{1}$

=>$0=\frac{-a}{1}$

=>$a=0$
and product of the zeros =$\alpha (-\alpha )=\frac{b}{1}$

=>$-{\alpha }^2=b$

=>$b=-{\alpha }^2$
=>$b$ has to be negative.
Hence, option $A$ is correct.