If (x+y)³-(x-y)³-6y(x²-y²)=ky² then k=
A. 1
B. 2
C. 4
D. 8
Answers
Given: (x + y)³ - (x - y)³ - 6y(x² - y²) = ky³
In order to factorise the given algebraic expression we use the following Identity - = (a + b)³ = a³ + b³ + 3a²b + 3ab² & (a - b)³ = a³ - b³ - 3a²b + 3ab² :
= x³ + y³ + 3x²y + 3xy² - (x³ - y³ - 3x²y + 3xy²) - 6y(x² - y²) = ky³
= x³ + y³ + 3x²y + 3xy² - x³ + y³ + 3x²y - 3xy² - 6x²y + 6y³ = ky³
= x³ - x³ + y³ + y³ + 6y³ + 3x²y + 3x²y - 6x²y + 3xy² - 3xy² = ky³
= 2y³ + 6y³ = ky²³
= 8y³ = ky³
On comparing :
= k = 8
Hence, the value of k is 8.
Option (D) 8 is correct.
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Identity to be used:-
x³ - y³ = (x - y) (x² + y² + xy)
=> (x + y)³ - (x - y)³ - 6y(x² - y²) = ky³
=> (x + y - x + y) ((x + y)² + (x - y)² + (x + y)(x - y)) = ky³
=> 2y (x² + y² + 2xy + x² + y² - 2xy + x² - y²) = ky³
=> 6x²y + 2y³ - 6x²y + 6y³ = ky³
=> 8y³ = ky³