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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