Question

The value of $P$ for which the equation $(P^3-3P^2+2P)x^2+(P^3-P)x+P^3+3P^2+2P=0$ has exactly one root at infinity is

• A. $1$
• B. $-1$
• C. $0$
• D. $2$

Answer 1

The correct option is D $2$

If $a{x}^2+bx+c=0$ has exactly one root at infinity, then it implies that $a=0$ and $b\ne 0$.
Given $(P^3-3P^2+2P)x^2+(P^3-P)x+P^3+3P^2+2P=0$
Comparing both, we get
$P^3-3P^2+2P=0$
$\Rightarrow P(P^2-3P+2)=0$
$\Rightarrow P(P^2-2P-P+2)=0$
$\Rightarrow P(P-1)(P-2)=0$
$\Rightarrow P=0,1,2\cdots \cdots \left(i\right)$

Also,
$P^3-P\ne 0$
$\Rightarrow P(P-1)(P+1)\ne 0$
$\Rightarrow P\ne 0,\pm 1\cdots \cdots \left(ii\right)$

From $\left(i\right)$ and $\left(ii\right)$, we get
$P=2$