Question

The minimum positive numerical value of $x$ for which the polynomial function $(x^2-1)(x^3-1)(x^4-1)$ is non-negative, is

Answer 1

Let $f\left(x\right)=(x^2-1)(x^3-1)(x^4-1)$
$=(x+1)(x-1)(x-1)(x^2+x+1)(x+1)(x-1)(x^2+1)$
$=(x^2+1)(x^2+x+1)(x+1{)}^2(x-1{)}^3$
$=(x^2+1)({(x+\frac{1}{2})}^2+\frac{3}{4})(x+1{)}^2(x-1{)}^3$
Given, $f\left(x\right)\ge 0$
$\Rightarrow (x-1{)}^3\ge 0\Rightarrow x\ge 1$