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# Factorisation using Identity

Algebra: Factorisation using Identity

Factorisation using Identity

(i) a 2 + 2ab + b 2 (a +b)2

(ii) a 2 2ab +b2 (a b)2

(iii) a2 b2 (a + b )(a b)

(iv) a 2 + b 2 +c 2 + 2ab + 2bc + 2ca (a + b +c)2

(v) a 3 + b 3 (a + b )(a 2 ab +b2 )

(vi) a 3 b 3 (a b )(a 2 +ab +b2 )

(vii) a 3 + b 3 +c 3 3abc (a + b +c )(a 2 + b 2 +c2 ab bc ca)

Note

(a + b )2 +(a − b)2 = 2(a2 + b2 ); a4 −b4 = (a2 +b2 )(a + b )(a −b)

(a + b )2 −(a −b)2 = 4ab ; a6b6 = (a +b)(a − b )(a2ab + b2 )(a2 +ab +b2 )

Progress Check

Example 3.25

Factorise the following:

(i) 9x2 + 12xy + 4y2 (ii) 25a2 10a +1 (iii) 36m2 49n2 (iv) x3 x (v) x4 − 16 (vi) x2 + 4y2 + 9z2 4xy +12yz 6xz

### Solution

(i) 9x2 + 12xy + 4y2 = (3x )2 + 2(3x )(2y ) +(2y)2 [ a 2 + 2ab + b2 = (a +b)2 ]

= (3x + 2y)2

(ii) 25a2 10a +1 = (5a )2 2(5a )(1) +12

= (5a −1)2    [ a 2 − 2ab + b2 = (ab)2 ]

(iii) 36m2 − 49n2. = (6m)2 -(7n)2

= (6m + 7n)(6m 7n)               [a 2 b2 = (a + b)(a b)]

(iv) x3 x = x (x2 1)

= x(x 2 12 )

= x (x +1)(x 1)

(v) x4 -16 = x4 -24         [a4 b4 =(a2 + b2)(a + b)(a − b)]

= (x 2 + 22 )(x2 22 )

= (x 2 + 4)(x + 2)(x 2)

(vi) x 2 + 4y 2 + 9z 2 4xy +12yz 6xz

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

= (x + 2y + 3z)2 or (x − 2y − 3z)2

### Example 3.26

Factorise the following:

(i) 27x 3 +125y3

(ii) 216m 3 − 343n3

(iii) 2x4 -16xy3

(iv) 8x3 + 27y3 + 64z3 72xyz

### Solution

(i) 27x3 +125y3

= (3x)3 +(5y)2     [ (a3 + b3 ) = (a + b)(a2 − ab + b2 ) ]

= (3x + 5y)((3x)2 (3x)(5y ) + (5y)2 )

= (3x + 5y)(9x2 15xy + 25y2 )

(ii) 216m 3 − 343n3 = (6m )3(7n)3         [a3b3 ) = (ab)(a 2+ ab + b2 )]

= (6m 7n) ((6m )2 + (6m)(7n ) + (7n)2 )

= (6m 7n)(36m 2 + 42mn + 49n2 )

(iii) 2x4 − 16xy3 = 2x (x 38y3 )

= 2x (x3 −(2y)3 )      [(a3b3 ) = (ab)(a2 + ab + b2)]

= 2x ((x2y )(x2 + (x)(2y) + (2y)2 ))

= 2x (x2y )(x2 + 2xy +4y2 )

(iv) 8x3 + 27y3 + 64z3 72xyz

= (2x)3 + (3y)3 + (4z)3 − 3(2x)(3y)(4z )

= (2x + 3y + 4z )(4x 2 + 9y 2 + 16z 2 6xy 12yz 8xz)

Thinking Corner

Check 15 divides the following

(i) 20173 + 20183

(ii) 20183 – 19733

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