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Algebraic Identities

Types of Polynomials Assignment / Homework Help
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)

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)

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)

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)

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)

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)

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)

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)

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)

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)


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