Lowest Common Multiple of Polynomials

How to find the lowest common multiple of polynomials?

To find the lowest common multiple (L.C.M.) of polynomials, we first find the factors of polynomials by the method of factorization and then adopt the same process of finding L.C.M.


Solved examples to find lowest common factor of polynomials:

1. Find the L.C.M. of 4a2 - 25b2 and 6a2 + 15ab.

Solution:

Factorizing 4a2 - 25b2 we get,

(2a)2 - (5b)2, by using the identity a2 - b2.

= (2a + 5b) (2a - 5b)


Also, factorizing 6a2 + 15ab by taking the common factor '3a', we get

= 3a(2a + 5b)

Therefore, the L.C.M. of 4a2 - 25b2 and 6a2 + 15ab is 3a(2a + 5b) (2a - 5b)



2. Find the L.C.M. of x2y2 - x2 and xy2 - 2xy - 3x.

Solution:

Factorizing x2y2 - x2 by taking the common factor 'x2' we get,

x2(y2 - 1)

Now by using the identity a2 - b2.

x2(y2 - 12)

= x2(y + 1) (y - 1)

Also, factorizing xy2 - 2xy - 3x by taking the common factor 'x' we get,

x(y2 - 2y - 3)

= x(y2 - 3y + y - 3)

= x[y(y - 3) + 1(y - 3)]

= x(y - 3) (y + 1)

Therefore, the L.C.M. of x2y2 - x2 and xy2 - 2xy - 3x is x2(y + 1) (y - 1) (y - 3).



3. Find the L.C.M. of x2 + xy, xz + yz and x2 + 2xy + y2.

Solution:

Factorizing x2 + xy by taking the common factor 'x', we get

x(x + y)

Factorizing xz + yz by taking the common factor 'z', we get

z(x + y)

Factorizing x2 + 2xy + y2 by using the identity (a + b)2, we get

= (x)2 + 2 (x) (y) + (y)2

= (x + y)2

= (x + y) (x + y)

Therefore, the L.C.M. of x2 + xy, xz + yz and x2 + 2xy + y2 is xz(x + y) (x + y).






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