How do you factor y=x3−2x2+x−2?

${\displaystyle (x-2)(x+i)(x-i)}$
Explanation:
factor the terms by grouping
${\displaystyle =x^2(x-2)+1(x-2)}$
take out the common factor ${\displaystyle (x-2)}$
${\displaystyle =(x-2)(x^2+1)}$
${\displaystyle x^2+1}$ can be factored using difference of squares
${\displaystyle a^2-b^2=(a-b)(a+b)}$
with ${\displaystyle a=x}$ and ${\displaystyle b=i\Rightarrow (i=\sqrt{-1})}$
${\displaystyle =(x-2)(x+i)(x-i)\Rightarrow \text{in factored form}}$

${\displaystyle y=(x-2)\times (x^2+1)}$
Explanation:
${\displaystyle x^3-2x^2}$ can be written as ${\displaystyle x^2\times (x-2)}$
${\displaystyle y=x^2\times (x-2)+(x-2)}$
or
${\displaystyle y=x^2\times (x-2)+1\times (x-2)}$
and bringing the common factor ${\displaystyle (x-2)}$ to the front, gives the factors:
${\displaystyle y=(x-2)\times (x^2+1)}$
solving the equation for ${\displaystyle y=0}$ gives the solution ${\displaystyle x=2}$ meaning that the graph of the equation crosses the x-axis at the point (2,0)
solving the equation for ${\displaystyle x=0}$ gives ${\displaystyle y=-2}$ , meaning that the graph of the equation crosses the y-axis at (0,-2)
graph ${\displaystyle \{x^3-2x^2+x-2[-10,10,-5,5]\}}$

$y=(x^2+1)(x-2)$
Explanation:
$y=x^3-2x^2+x-2$
We will use a factoring method called grouping. If we group the first and the third and the second and the fourth together, each grout has its own greatest common factor:
$y=x(x^2+1)-2(x^2+1)$
Now we see another CF between the two terms : $x^2+1$ . We can factor it out.
$y=(x^2+1)(x-2)$