Question

The number of distinct real roots of the cubic polynomial equation $x^3-3x^2+3x-1=0$ is

• A. $1$
• B. $2$
• C. $0$
• D. $3$

Answer 1

The correct option is A $1$

Given: $f\left(x\right)=x^3-3x^2+3x-1=0$

Let's substitute $x=1$ in $f\left(x\right)$, we get:
$f\left(1\right)=1^3-3(1{)}^2+3(1)-1=0$

Hence, we know that
$x=1$ is a root of $f\left(x\right)$ or $(x-1)$ is its factor.
Apply divison algorithm,
$\therefore f\left(x\right)=(x-1)(x^2-2x+1)$
$f\left(x\right)=(x-1)(x^2+(-1-1)x+(-1\left)\right(-1\left)\right)$
$\Rightarrow f\left(x\right)=(x-1)(x-1)(x-1)$
Roots of $f\left(x\right)=0$ are $x=1,1,1$
So number of distinct real roots is 1.