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)