Example x^{2}  1 ; x^{2} + 3x + 2 ; x^{2} + 4x + 3
Solution: x^{2}  1 = ( x  1 ) ( x + 1 )
x^{2} + 3x + 2 = ( x + 2 ) ( x + 1 )
x^{2} + 4x + 3 = ( x + 3 ) ( x + 1 )
\ H.C.F = ( x + 1 ) and
L.C.M = ( x  1 ) ( x + 1 ) ( x + 2 ) ( x + 3 )
H.C.F. and L.C.M. by long division method.
If the factors of a given polynomial cannot be
found out easily and degrees are not same, then we may employ another
method known as long division method for finding out H.C.F. and
L.C.M.
1) Take higher degree polynomial and divide
it by the other polynomial of lower degree till we get zero remainder.
Then the divisor is the H.C.F. ( G.C.D.) of the two polynomials.
L.C.M. =
Example Find G.C.D. and L.C.M. of x^{2}
+ x + 1 and x^{4} + x^{2} + 1
Soltuion: Dividing the higher degree polynomial
x^{4} + x^{2} + 1 by the lower degree polynomial x^{2} + x + 1 as :
Therefore, G.C.D.= x^{2} + x + 1
and L.C.M. =
= x^{4} + x^{2} + 1
2) If the remainder is not zero, then divide the divisor
by remainder till the latest remainder is zero. Then the second
divisor ( i.e. the first remainder ) is G.C.D. If > again remainder
is not zero, continue the process till you get remainder zero. The
last divisor is the G.C.D. and L.C.M.
=
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