Question

State true or false:

If two of the zeros of a cubic polynomial are zero, then it does not have linear and constant terms.

• A. True
• B. False

Answer 1

The correct option is B False

Considering a cubic polynomial,
$p\left(x\right)=a{x}^3+b{x}^2+cx+d$
Let $\alpha ,\beta ,\gamma$ be the zeros of the polynomial.
Then,two of these must be zero.
Let $\alpha =\beta =0$.
Since,$\alpha =0$ is a zero of the polynomial,so $p\left(0\right)=0$
$=>a{\left(0\right)}^3+b(0{)}^2+c(0)+d=0$
$=>0+0+0+d=0$
$=>d=0$
Therefore $p\left(x\right)$ reduces to
$p\left(x\right)=a{x}^3+b{x}^2+cx=x(a{x}^2+bx+c)$
For the zeros we have $p\left(x\right)=0$
$=>x(a{x}^2+bx+c)=0$
$=>x=0$or $(a{x}^2+bx+c)=0$ (i)
Also given that $\beta =0$ is the zero of polynomial.
From equation (i) we get,
$=>a{\left(0\right)}^2+b\left(0\right)+c=0$
$=>c=0$
Hence $c=0$ and $d=0$
Therefore $p\left(x\right)$ does not have linear and constant term.
Hence, the given statement is true.