Question

$f\left(x\right)=\frac{x^3-6x^2+11x-6}{x-3}\forall x\in R$ is a polynomial.

• A. True
• B. False

Answer 1

The correct option is B False

Given that $f\left(x\right)=\frac{x^3-6x^2+11x-6}{x-3}\forall x\in R$
$\Rightarrow f\left(x\right)=\frac{x^3-5x^2-x^2+6x+5x-6}{x-3}$
$\Rightarrow f\left(x\right)=\frac{x^3-x^2+6x-6-5x^2+5x}{x-3}$
$\Rightarrow f\left(x\right)=\frac{x^2(x-1)+6(x-1)-5x(x-1)}{x-3}$
$\Rightarrow f\left(x\right)=\frac{(x-1)(x^2-5x+6)}{x-3}$
$\Rightarrow f\left(x\right)=\frac{(x-1)(x-2)(x-3)}{x-3}$
$\Rightarrow x\ne 3$
$f\left(x\right)$ is undefined at $x=3$
A polynomial function has to be defined $\forall x\in R$
$\therefore f\left(x\right)$ is not polynomial function for all real $x$