Question

A monic quadratic trionmoial $p\left(x\right)$ is such that $P\left(x\right)=0$ and $P\left(P\right(P\left(x\right)\left)\right)=0$ have a common root then :

• A. $P\left(0\right)P\left(1\right)>0$
• B. $P\left(0\right)P\left(1\right)<0$
• C. $P\left(0\right)P\left(1\right)=0$
• D. $Noneofthese$

Answer 1

Answer: (C)

$P\left(0\right)P\left(1\right)=0$

We have,

$x^2+ax+b$

$p\left(p\left(p\left(x\right)\right)\right)=0$

$p\left(p\left(x\right)\right)=0$

$p\left(p\left(x\right)\right)=a=0$

$p\left(x\right)=a_1.a_2$

$x=a_{11}{a}_{12}{a}_{21}{a}_{22}$

$\therefore p\left(0\right)=0$

$\therefore p\left(0\right).p\left(1\right)=0$

Hence, this is the answer.