Question

The number of the distinct zeroes of the polynomial $f\left(x\right)=x(x-4{)}^3(x-3{)}^2(x-1)$ is

Answer 1

Given polynomial $f\left(x\right)=x(x-4{)}^3(x-3{)}^2(x-1)$
For zeroes of a polynomial
$f\left(x\right)=0\Rightarrow x(x-4{)}^3(x-3{)}^2(x-1)=0\Rightarrow x=0,(x-4{)}^3=0,(x-3{)}^2=0,(x-1)=0\Rightarrow x=0,4,4,4,3,3,1$
$\therefore$ The distinct zeroes of the polynomial are $x=0,1,3,4$
Hence, the number of distinct zeroes of the polynomial is $4$.