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Question

Find the LCM of 8x3y3,ab(4x2+2xy+y2),bc(4x2y2)

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Solution

We know that LCM is the least common multiple.

Factorise 8x3y3 as follows:

8x3y3=(2x)3y3=(2xy)[(2x)2+y2+(2x×y)]=(2xy)(4x2+y2+2xy)
(using identity a3b3=(ab)(a2+b2+ab))

Now, factorise bc(4x2y2) as follows:

bc(4x2y2)=bc[(2x)2y2)]=bc(2x+y)(2xy) (using identity a2b2=(a+b)(ab))

Therefore, the least common multiple between the polynomials 8x3y3, ab(4x2+2xy+y2)and bc(4x2y2) is:

LCM=abc(2xy)(2x+y)(4x2+2xy+y2)=abc(2x+y)(8x3y3)
(using identity a3b3=(ab)(a2+b2+ab))

Hence, the LCM is abc(2x+y)(8x3y3).

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