Q4 E of 42 Page 136

Let’s find the L.C.M. of the following algebraic expressions:

x4 + x2y2 + y4, x3y + y4, (x2-xy)3


factors of x4 + x2y2 + y4:

Add and subtract x2y2 to get,


x4 + x2y2 + y4 = x4 + x2y2 + y4 + x2y2 - x2y2


= x4 + 2x2y2 + y4 - x2y2


As we know (a + b)2 = a2 + b2 + 2ab,


So,
x4 + x2y2 + y4 = (x2 + y2)2 - x2y2


= (x2 + y2)2 – (xy)2


Apply the formula a2 – b2 = (a + b) (a-b) in (x2 + y2)2 – (xy)2to get,


= (x2 + y2-xy) (x2 + y2+xy)


Now factorise x3y + y4,


= y (x3 + y3)


Apply a3 + b3 = (a + b)(a2 + b2 – ab) in (x3 + y3)


= y (x+y)(x2 + y2 – xy)


Now factorise (x2-xy)3,


= (x2-xy)2(x2-xy)


Apply (a-b)2 = a2 + b2 – 2ab in (x2-xy)2 to get,


= [(x2)2 + (xy)2 – 2x2xy][x(x-y)]


= x2 [x2 + y2 – 2xy][x(x-y)]


As we know (a - b)2 = a2 + b2 - 2ab in ,


= x3 (x-y)2 (x-y)


= x3 (x-y)3


So the LCM of x4 + x2y2 + y4, x3y + y4, (x2-xy)3 is:


x3y (x-y)3(x+y)(x2 + y2 – xy) (x2 + y2+xy)


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