CLASS-8ALGEBRAIC EXPANSION- CUBES OF BINOMIAL

CUBES OF BINOMIAL –

Now, let us find the cubes of the sum and the difference of the two terms x & y, in other words, let us find (x+y)ᶟ and  (x-y)ᶟ

A)   (x+y)ᶟ = xᶟ + 3x²y + 3xy² + yᶟ

proof -  (x+y)ᶟ = (x+y)².(x+y) = (x²+ 2xy + y²)(x+y)

=  {(x²+ 2xy + y²).x + (x²+ 2xy + y²).y}

=  xᶟ + 2x²y + xy² + x²y + 2xy² + yᶟ

=  xᶟ + 3x²y + 3xy² + yᶟ         (Proven)

So,  (x+y)ᶟ = xᶟ + 3x²y + 3xy²+ yᶟ

There are some example are given below for your better understanding –

Example.1) Find the value of (2a + 3)ᶟ = ?

Ans.)      (2a + 3)ᶟ

=  (2a)ᶟ + 3X (2a)²X 3 + 3 X 2a X 3² + 3ᶟ

=  8aᶟ + 36a² + 54a + 27            (Ans.)

Example.2)  Find the value of (3a + 2b)ᶟ = ?

Ans.)      (3a - 2b)ᶟ

=  (3a)ᶟ - 3 X (3a)² X 2b + 3 X 3a X (2b)² - (2b)ᶟ

=  27aᶟ - 54a²b + 36ab² - 8bᶟ          (Ans.)

Corollaries –

Two corollaries follow from this expansion.

1)   (a+b)ᶟ =  aᶟ + 3a²b + 3ab² + bᶟ =  aᶟ + bᶟ + 3ab(a+b)

So,   (a+b)ᶟ = aᶟ + bᶟ + 3ab(a+b)

2)   (a+b)ᶟ - 3ab(a+b) = aᶟ + bᶟ + 3ab(a+b) – 3ab(a+b) =  aᶟ + bᶟ

So,  aᶟ + bᶟ = (a+b)ᶟ - 3ab(a+b)

B)   (x-y)ᶟ = xᶟ - 3x²y + 3xy² - yᶟ

proof -    (x-y)ᶟ =  (x-y)².(x-y)

=  (x²- 2xy + y²)(x-y)

=  {(x²- 2xy + y²).x - (x²- 2xy + y²).y}

=  xᶟ - 2x²y + xy² - x²y + 2xy² - yᶟ

=  xᶟ - 3x²y + 3xy²- yᶟ        (Proven)

So, (x-y)ᶟ = xᶟ - 3x²y + 3xy²- yᶟ

Corollaries –

The corollaries that follow are –

1)   (a – b)ᶟ = aᶟ - bᶟ - 3ab(a - b)

2)   (a – b)ᶟ + 3ab(a-b) = aᶟ - bᶟ - 3ab(a-b) + 3ab(a-b) =  aᶟ - bᶟ

So,  aᶟ - bᶟ = (a – b)ᶟ + 3ab(a-b)