Solve the polynomial inequality x2−2x+1>0 and graph the solution set on a real number line....

Step 1
The given polynomial inequality is
${\displaystyle x^2-2x+1>0}$
To solve the inequality, find its critical points by equating it to zero.
${\displaystyle x^2-2x+1=0}$
${\displaystyle x^2-x-x+1=0}$
${\displaystyle x(x-1)-1(x-1)=0}$
${\displaystyle (x-1)(x-1)=0}$
${\displaystyle x=1,1}$
Step 2
Now, make the sign analysis on Number line to determine about the inequality.
If ${\displaystyle x<1}$, then for the given polynomial the result is always positive and even if ${\displaystyle x>1}$, then also for the given polynomial the result is positive. Therefore, for all values of x except 1 ,the polynomial is positive.

Answer: Hence, the interval notation for the given polynomial inequality is
${\displaystyle (-\mathrm{\infty },1)\cup (1,\mathrm{\infty })}$