Relationship between Least Common Multiple (LCM) of Polynomials and Greatest Common Divisor (GCD) or Highest Common Factor (HCF).

**Relationship between LCM and GCD**

Let us consider two
numbers 12 and 18.

We observe that, LCM
(12, 18) = 36, GCD (12, 18) = 6 .

Now, LCM (12,18) ×
GCD(12,18) = 36 ×6 = 216 = 12 ×18

Thus LCM × GCD is equal
to the product of two given numbers.

Similarly, the product
of two polynomials is the product of their LCM and GCD,

That is, *f* (*x*)
× *g* (*x*) = *LCM* [ *f* (*x* ), *g*(*x* )]
×*GCD*[ *f* (*x*), *g* (*x*)]

Consider * f*(*x*)
= 12(*x* ^{2} −*y*^{2} ) and *g* (*x* ) =
8(*x* ^{3} −*y*^{3} )

Now
*f*(*x*) = 12(*x* ^{2} −*y*^{2}
) = 2^{2} × 3 ×(*x* + *y* )(*x* −*y*)
...(1)

And
*g*(*x*) = 8(*x* ^{3} −*y*^{3}
) = 2 ^{3} ×(*x* − *y* )(*x* ^{2} + *xy* + *y*^{2}
) ...(2)

From (1) and (2) we get,

LCM[*f*(*x*), *g*(*x*)]
= 2 ^{3} ×3 × (*x* + *y*)(*x* − *y* )(*x* ^{2}
+ *xy* + *y*^{2} )

= 24 ×(*x* ^{2}
− *y* ^{2} )(*x* ^{2} + *xy* + *y*^{2}
)

*GCD f *(*x *),* g*(*x*)
= 2^{2} ×(*x* − *y* ) = 4(*x* −*y*)

*LCM *×*GCD *= 24 ×4 × (*x*
^{2} −*y*^{2} )× (*x* ^{2} + *xy* + *y*
^{2} )×(*x* −*y*)

*LCM *×*GCD *= 96(*x*
^{3} −*y*^{3} )(*x* ^{2} −*y*^{2}
) ...(3)

product of *f*(*x*)
and *g*(*x*) = 12(*x* ^{2} −*y*^{2} )× 8(*x*
^{3} −*y*^{3} )

= 96(*x* ^{2}
−*y*^{2} )(* x* ^{3} −*y*^{3} )
...(4)

From (3) and (4) we obtain LCM × GCD = *f*
(*x* )×*g* (*x* )

Tags : Illustration | Algebra , 10th Mathematics : Algebra

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10th Mathematics : Algebra : Relationship between LCM and GCD | Illustration | Algebra

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