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

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

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Answer 1 · 2018-03-11T15:05:14

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

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! :)

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

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