Question

$f\left(x\right)=\frac{x^3+9x^2+23x+15}{x+1}\forall x\in R$ is not a polynomial.

• A. False
• B. True

Answer 1

The correct option is B True

Given that $f\left(x\right)=\frac{x^3+9x^2+23x+15}{x+1}\forall x\in R$
$\Rightarrow f\left(x\right)=\frac{x^3+x^2+8x^2+8x+15x+15}{x+1}$
$\Rightarrow f\left(x\right)=\frac{x^2(x+1)+8x(x+1)+15(x+1)}{x+1}$
$\Rightarrow f\left(x\right)=\frac{(x+1)(x^2+8x+15)}{x+1}$
$\Rightarrow f\left(x\right)=\frac{(x+1)(x+3)(x+5)}{x+1}$
$\Rightarrow x\ne -1$
$f\left(x\right)$ is undefined at $x=-1$
A polynomial function has to be defined $\forall x\in R$
$\therefore f\left(x\right)$ is not polynomial function for all real $x$