Question

If $p\left(x\right)$ be a polynomial satisfying the identity $p\left(x^2\right)+2x^2+10x=2xp(x+1)+3,$ then function $p\left(x\right)$ is

• A. One-one
• B. Many-one
• C. Periodic
• D. Even

Answer 1

Answer: (A)

One-one

Degree of expression on LHS = degree of expression on RHS

LHS = RHS

$\Rightarrow 2n=n+1\Rightarrow n=1$

$\therefore$ Let $p\left(x\right)=ax+b$

As $p\left(x^2\right)+2x^2+10x=2xp(x+1)+3$

$\Rightarrow a{x}^2+b+2x^2+10x=2x(ax+a+b)+3$

$\Rightarrow a+2=2a,10=2a+2b$ and $b=3$

$\Rightarrow a=2,b=3$

$\therefore p\left(x\right)=2x+3$

Hence, it is one-one.