Question

How do you factor `x^3 - 2x^2 - x + 2` by grouping?

Answer 1

`color(magenta)(=(x-1)(x+1)(x-2)`

Explanation:

`x^3-2x^2-x+2`

`=x^3-x-2x^2+2`

Grouping the `1^(st` `2` terms together and the `2^(nd` `2` together:

`=x(x^2-1)-2(x^2-1)`

`=(x^2-1)(x-2)`

Using the identity: `a^2-b^2=(a+b)(a-b)`

`=(x^2-1^2)(x-2)`

`color(magenta)(=(x-1)(x+1)(x-2)`

`color(red)("Alternate Method:-"`

The above method is an easy on to solve this question. For factorizing other cubic polynomials, the following method can be used:

First, by trial and error method, you can find one factor as follows:

`x^3-2x^2-x+2`

When replacing 1,

`=>1^3-2xx1^2-1+2`

`=>cancel1-cancel2-cancel1+cancel2`

`=>0`

So, we get `(x-1)` as factor.

Then by long division, divide `(x-1)` by `(x^3-2x^2-x+2)`

You get`=>(x-1)(x^2-x-2)`

Then you have to factorize it by splitting the middle term method.

`=>(x-1)[x^2+x-2x-2]`

`=>(x-1)[x(x+1)-2(x+1)]`

`color(magenta)(=>(x-1)(x+1)(x-2)`

~Hope this helps! :)