Factorisation Using Square Identities


 
 
Concept Explanation
 

Factorisation Using Square Identities

Factorisation Using the Difference of the Squares:

To factorize binomial expressions expressible as the difference of two squares, we use the following identity:

                            a^{2}-b^{2}=(a+b)(a-b)

Illustration: Factorise the following 

                 (i);x^{4}-y^{4}          =(x^{2})^{2}-(y^{2})^{2}

                                                  =(x^{2}-y^{2})(x^{2}+y^{2})               [Using:;;a^{2}-b^{2}=(a-b)(a+b)]

                                                  =(x-y)(x+y)(x^{2}+y^{2})  [Using:;;a^{2}-b^{2}=(a-b)(a+b)]

                large (ii);x^{4}-(y+z)^{4}    =(x^{2})^{2}-left { (y+z)^{2} right }^{2}

                                                         large =left { x^{2}-(y+z)^{2} right }left { x^{2}+(y+z)^{2} right }

                                                       large =left { x-(y+z) right }left { x+(y+z) right }left { x^{2}+(y+z)^{2} right }

            large (iii);2x-32x^{5}        large =2x(1-16x^{4})

                                                     large =2xleft { 1^{2}-(4x^{2})^{2} right }

                                                     large =2x(1+4x^{2})(1-4x^{2})

                                                    large =2x(1+4x^{2})left { 1-(2x)^{2} right }

                                                    large =2x(1+4x^{2})(1-2x)(1+2x)

Factorization Using Perfect Square Identities:

In order to factorize algebraic expressions expressible as a perfect square, we use yhr following expressions.

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

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

Illustration: Factorise the following

                  (i);4x^{2}+12xy+9y^{2}   =(2x)^{2}+2times 2xtimes 3y+(3y)^{2}

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

                                                                      =(2x+3y)(2x+3y)

                (ii);x^{4}+10x^{2}y^{2}+25y^{4}   =(x^{2})^{2}-2times x^{2}times 5y^{2}+(5y^{2})^{2}

                                                                     =(x^{2}-5y^{2})^{2}

                                                                     =(x^{2}-5y^{2})(x^{2}-5y^{2})        

Sample Questions
(More Questions for each concept available in Login)
Question : 1

Factorize using identities :    -16x^{2}+64x^{4}

Right Option : B
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Question : 2

Which of the following expressions is the factor of large 16x^2-4 ?

Right Option : A
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Question : 3

If  (x^2+3x+5)(x^2-3x+5)=m^2-n^2,  then m and n is

Right Option : B
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