Question

How do you factor `y= x^3-2x^2+x-2`?

Answer 1

`(x-2)(x+i)(x-i)`

Explanation:

`"factor the terms "color(blue)"by grouping"`

`=color(red)(x^2)(x-2)color(red)(+1)(x-2)`

`"take out the "color(blue)"common factor "(x-2)`

`=(x-2)(color(red)(x^2+1))`

`x^2+1" can be factored using "color(blue)"difference of squares"`

`a^2-b^2=(a-b)(a+b)`

`"with "a=x" and "b=ito(i=sqrt-1)`

`=(x-2)(x+i)(x-i)larrcolor(red)"in factored form"`

Answer 2

`y=(x-2) xx (x^2+ 1)`

Explanation:

`x^3 -2x^2` can be written as
`x^2xx(x-2)`, so we can re-write the original equation as

`y=x^2xx(x-2) + (x-2)`
or
`y=x^2xx(x-2) + 1xx(x-2)`

and bringing the common factor `(x-2)` to the front, gives the factors:

`y=(x-2) xx (x^2+ 1)`

solving the equation for `y=0`, gives the solution `x=2`, meaning that the graph of the equation crosses the x-axis at the point (2,0)

solving the equation for `x=0` gives `y=-2`, meaning that the graph of the equation crosses the y-axis at (0,-2)

graph{x^3-2x^2+x-2 [-10, 10, -5, 5]}

Answer 3

`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)`