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Algebra I: Relationship between H.C.F. and L.C.M.
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Relationship between H.C.F. and L.C.M. of Two Polynomials

If f(x) and g(x) are two polynomials, then 
           f(x). g(x) = {HCF of f(x) and g(x)} x {L.C.M of f(x) and g(x)}.
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Example: Find the L.C.M of the following pairs of polynomials with the help of their H.C.F.:
3 (x2 - 7x + 12) and 24(x2 - 9x + 20)
Solution:
Given,
p(x)
= 3 (x2 - 7x + 12)
= 3(x - 3)(x - 4)
and, q(x)
= 24(x2 - 9x + 20)
= 23 x 3 x (x - 4)(x - 5)

Clearly, H.C.F. of p(x) and q(x) is 3 (x - 4). 

Therefore,
L.C.M. of p(x) and q(x)
= [p(x). q(x)] / H.C.F. of p(x) and q(x
= { [3 (x - 3) (x - 4)1 x [23 x 3 x (x - 4) (x- 5)]}/ 3 (x - 4)
= (x - 3) x 23 x 3 x (x - 4) (x - 5)
= 24(x - 3)(x - 4)(x - 5)
 
   
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