We know that LCM is the least common multiple.
Factorise 8x3−y3 as follows:
8x3−y3=(2x)3−y3=(2x−y)[(2x)2+y2+(2x×y)]=(2x−y)(4x2+y2+2xy)
(using identity a3−b3=(a−b)(a2+b2+ab))
Now, factorise bc(4x2−y2) as follows:
bc(4x2−y2)=bc[(2x)2−y2)]=bc(2x+y)(2x−y) (using identity a2−b2=(a+b)(a−b))
Therefore, the least common multiple between the polynomials 8x3−y3, ab(4x2+2xy+y2)and bc(4x2−y2) is:
LCM=abc(2x−y)(2x+y)(4x2+2xy+y2)=abc(2x+y)(8x3−y3)
(using identity a3−b3=(a−b)(a2+b2+ab))
Hence, the LCM is abc(2x+y)(8x3−y3).