    # Algebraic Identities 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

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)

Online Algebraic Identity Help:

If you are stuck with a Algebraic Identity Homework problem and need help, we have excellent tutors who can provide you with Homework Help. Our tutors who provide Algebraic Identity help are highly qualified. Our tutors have many years of industry experience and have had years of experience providing Algebraic Identity Homework Help. Please do send us the Algebraic Identity problems on which you need Help and we will forward then to our tutors for review.

Online Tutor Algebraic Identities:

We have the best tutors in math in the industry. Our tutors can break down a complex Algebraic Identities problem into its sub parts and explain to you in detail how each step is performed. This approach of breaking down a problem has been appreciated by majority of our students for learning Algebraic Identities concepts. You will get one-to-one personalized attention through our online tutoring which will make learning fun and easy. Our tutors are highly qualified and hold advanced degrees. Please do send us a request for Algebraic Identities tutoring and experience the quality yourself.

Other topics under Number Theory:       • 