Factorisation Using Cubic Identities


 
 
Concept Explanation
 

Factorisation Using Cubic Identities

Factorization of Algebraic Expressions Expressible as a Perfect Cube:

In order to factorize algebraic expressions expressible as a perfect cube, we use the following expressions.

 large (i);a^{3}+3a^2b+3ab^2+b^{3}=(a+b)^{3}=(a+b)(a+b)(a+b)

 large (ii);a^{3}-3a^2b+3ab^2+b^{3}=(a-b)^{3}=(a-b)(a-b)(a-b)

Illustration: Factorise the following

large (i);8x^3+36x^2y+54xy^2+27y^3  

large =(2x)^{3}+3times (2x)^2times 3y+3times 2xtimes (3y)^2+(3y)^{3}

 large =(2x+3y)^{3}    large =(2x+3y)(2x+3y)(2x+3y)

Factorization of Algebraic Expressions using Conditional Identity:

In order to factorize algebraic expressions expressible as the following expressions.

 large ;a^{3}+b^3+c^3-3abc= (a+b+c)(a^2+b^2+c^2-ab-bc-ca)

Illustration: Factorise the following

 large 8x^3+y^3+27z^3-18xyz  

large =(2x)^{3}+y^3+(3z)^3-3 times (2x)times (y) times (3z)

=(2x+y+3z)((2x)^{2}+y^2+(3z)^2-2x times (y)- (y)times (3z)-(3z) times (2x))

large =(2x+y+3z)(4x^2+y^2+9z^2-2xy- 3yz -6xz)

Sample Questions
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Question : 1

The factors of  8a^3+b^3-6ab+1: : are

Right Option : C
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Explanation
Question : 2

One of the factor of

a^{3}+8b^{2}-64c^{3}+24abc ;;is:

Right Option : A
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Explanation
Question : 3

a^1^2-1  can be factorised as :

Right Option : D
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Explanation
 
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