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ALGEBRAIC IDENTITY
DEFINITION:
An algebraic equation which is true for all values of the variables occurring in the relation is known as an algebraic identity.
IMPORTANT IDENTITIES:
There are 8 important algebraic identities which are given below:
POLYNOMIALS OF DIFFERENT DEGREES:
Identity I:
(x + y)2 = x2 + 2xy + y2
Identity II:
(x - y)2 = x2 - 2xy + y2
Identity III:
x2 - y2 = (x+ y)(x - y)
Identity IV:
(x + a)(x + b) = x2 + (a + b)x + ab
Identity V:
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
Proof:
Let x + y = k then,
(x + y + z)2 = (k + z)2
= k2 + 2kz + z2 (Using identity I)
= (x + y)2 + 2(x + y)z + z2
= x2 + 2xy + y2 + 2 xz + 2yz + z2
= x2 + y2 + z2 + 2xy + 2yz + 2zx (proved)
Let x + y = k then,
(x + y + z)2 = (k + z)2
= k2 + 2kz + z2 (Using identity I)
= (x + y)2 + 2(x + y)z + z2
= x2 + 2xy + y2 + 2 xz + 2yz + z2
= x2 + y2 + z2 + 2xy + 2yz + 2zx (proved)
Identity VI:
(x + y)3 = x3 + y3 + 3xy(x + y)
Identity VII:
(x - y)3 = x3 - y3 - 3xy(x - y)
Identity VIII:
x3 + y3 + z3 - 3xyz = (x + y + z)(x2 +
y2 + z2 - xy - yz - zx)
Proof:
R.H.S.
= (x + y + z)(x2 + y2 + z2 - xy - yz - zx)
= x(x2 + y2 + z2 - xy - yz - zx) + y(x2 + y2 + z2 - xy - yz - zx)
+ z(x2 + y2 + z2 - xy - yz - zx)
= x3 + xy2 + xz2 - x2y - xyz - zx2 + yx2 + y3 + yz2- xy2 - y2z - xyz
+ zx2 + zy2 + z3 - xyz - yz2 - xz2
= x3 + y3 + z3 - 3xyz = L.H.S. (proved)
R.H.S.
= (x + y + z)(x2 + y2 + z2 - xy - yz - zx)
= x(x2 + y2 + z2 - xy - yz - zx) + y(x2 + y2 + z2 - xy - yz - zx)
+ z(x2 + y2 + z2 - xy - yz - zx)
= x3 + xy2 + xz2 - x2y - xyz - zx2 + yx2 + y3 + yz2- xy2 - y2z - xyz
+ zx2 + zy2 + z3 - xyz - yz2 - xz2
= x3 + y3 + z3 - 3xyz = L.H.S. (proved)
Example 1:
Evaluate 95 x 96.
Solution:
95 x 96 = (90 + 5) x (90 + 6)
= (90)2 + (5 + 6)(90) + (5)(6) (Using identity IV)
= 8100 + 990 + 30 = 9120 (Answer)
Evaluate 95 x 96.
Solution:
95 x 96 = (90 + 5) x (90 + 6)
= (90)2 + (5 + 6)(90) + (5)(6) (Using identity IV)
= 8100 + 990 + 30 = 9120 (Answer)
Example 2:
Evaluate 104 x 96.
Solution:
104 x 96 = (100 + 4) x (100 - 4)
= (100)2 - (4)2 (Using identity III)
= 10000 - 16
= 9984 (Answer)
Evaluate 104 x 96.
Solution:
104 x 96 = (100 + 4) x (100 - 4)
= (100)2 - (4)2 (Using identity III)
= 10000 - 16
= 9984 (Answer)
Example 3:
Factorise 4x2 + 2xy + y2.
Solution:
4x2 + 2xy + y2 = (2x)2 + 2(2x)(y) + (y)2
= (2x + y)2 (Using identity I)
= (2x + y)(2x + y) (Answer)
Factorise 4x2 + 2xy + y2.
Solution:
4x2 + 2xy + y2 = (2x)2 + 2(2x)(y) + (y)2
= (2x + y)2 (Using identity I)
= (2x + y)(2x + y) (Answer)
Example 4:
Factorise 9x2 - 6xy + y2.
Solution:
9x2 - 6xy + y2 = (3x)2 - 2(3x)(y) + (y)2
= (3x - y)2 (Using identity II)
= (3x - y) (3x - y) (Answer)
Factorise 9x2 - 6xy + y2.
Solution:
9x2 - 6xy + y2 = (3x)2 - 2(3x)(y) + (y)2
= (3x - y)2 (Using identity II)
= (3x - y) (3x - y) (Answer)
Example 5:
Expand (3x - 7y - z)2.
Solution:
(3x - 7y - z)2 = {3x+(-7y)+(-z)}2 (Using identity V)
= (3x)2 + (-7y)2 + (-z)2 + 2(3x)(-7y) + 2(-7y)(-z) + 2(-z)(3x)
= 9x2 + 49y2 + z2 - 42xy + 14yz - 6zx (Answer)
Expand (3x - 7y - z)2.
Solution:
(3x - 7y - z)2 = {3x+(-7y)+(-z)}2 (Using identity V)
= (3x)2 + (-7y)2 + (-z)2 + 2(3x)(-7y) + 2(-7y)(-z) + 2(-z)(3x)
= 9x2 + 49y2 + z2 - 42xy + 14yz - 6zx (Answer)
Example 6:
Expand (2x + 1)3.
Solution:
(2x + 1)3 = (2x)3 + (1)3+ 3(2x)(1)(2x + 1) (Using identity VI)
= 8x3 + 1 + 6x(2x + 1)
= 8x3 + 12x2 + 6x + 1 (Answer)
Expand (2x + 1)3.
Solution:
(2x + 1)3 = (2x)3 + (1)3+ 3(2x)(1)(2x + 1) (Using identity VI)
= 8x3 + 1 + 6x(2x + 1)
= 8x3 + 12x2 + 6x + 1 (Answer)
Example 7:
Expand (2x - 3y)3.
Solution:
(2x - 3y)3 = (2x)3 - (3y)3 - 3(2x)(3y)(2x - 3y) (Using identity IV)
= 8x3 - 27y3 - 18xy(2x - 3y)
= 8x3 - 27y3 - 36x2y + 54xy2 (Answer)
Expand (2x - 3y)3.
Solution:
(2x - 3y)3 = (2x)3 - (3y)3 - 3(2x)(3y)(2x - 3y) (Using identity IV)
= 8x3 - 27y3 - 18xy(2x - 3y)
= 8x3 - 27y3 - 36x2y + 54xy2 (Answer)
Example 8:
Factorise 27x3 + y3 + z3 - 9xyz.
Solution:
27x3 + y3 + z3 - 9xyz = (3x)3 + (y)3 + (z)3 - 3(3x)(y)(z)
= (3x + y + z){(3x)2 + (y)2 + (z)2 - (3x)(y) - (y)(z) - (z)(3x)} (Using identity VIII)
= (3x + y + z)(9x2 + y2 + z2 - 3xy - yz - 3zx) (Answer)
Factorise 27x3 + y3 + z3 - 9xyz.
Solution:
27x3 + y3 + z3 - 9xyz = (3x)3 + (y)3 + (z)3 - 3(3x)(y)(z)
= (3x + y + z){(3x)2 + (y)2 + (z)2 - (3x)(y) - (y)(z) - (z)(3x)} (Using identity VIII)
= (3x + y + z)(9x2 + y2 + z2 - 3xy - yz - 3zx) (Answer)
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