Question

factories X cube _2xsquare_x+2
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Answer 1

$\large\underline{\sf{Solution-}}$

Concept Used :-

1. Method of Common Factors

• In this method, we have to write the irreducible factors of all the terms

• Then find the common factors amongst all the irreducible factors.

• The required factor form is the product of the common term we had chosen and the left over terms.

2. Factorisation by Regrouping Terms

• Sometimes it happens that there is no common term in the expressions then

• We have to make the groups of the terms.

• Then choose the common factor among these groups.

• Find the common binomial factor and it will give the required factors.

Let's solve the problem now!!

$\rm :\longmapsto\: {x}^{3} - {2x}^{2} - x + 2$

$\rm :\longmapsto\: = \: ( {x}^{3} - {2x}^{2} ) + ( - x + 2)$

$\rm :\longmapsto\: = \: {x}^{2} (x - 2) - 1(x - 2)$

$\rm :\longmapsto\: = \: (x - 2)( {x}^{2} - 1)$

$\rm :\longmapsto\: = \: (x - 2)( {x}^{2} - {1}^{2} )$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \sf{ \because \: {x}^{2} - {y}^{2} = (x - y)(x + y)}}$

$\rm :\longmapsto\: = \: (x - 2)(x - 1)(x + 1)$

$\rm :\implies\: {x}^{3} - {2x}^{2} - x + 2 = (x - 2)(x - 1)(x + 1)$

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)