The Least Common Multiple of two or more algebraic expressions is the expression of lowest degree (or power) such that the expressions exactly divide it.

**Least Common Multiple (LCM) of Polynomials**

The Least Common Multiple of two or more
algebraic expressions is the expression of lowest degree (or power) such that
the expressions exactly divide it.

Consider the following
simple expressions *a*^{3}*b*^{2}, *a*^{2}*b*^{3}
.

For these expressions *LCM*
= *a*^{3}*b*^{3}.

To find LCM by
factorization method

(i) Each expression is
first resolved into its factors.

(ii) The highest power
of the factors will be the LCM.

(iii) If the expressions
have numerical coefficients, find their LCM.

(iv) The product of the
LCM of factors and coefficient is the required LCM.

Find the LCM of the
following

(i) 8*x*^{4}*y*^{2},
48*x*^{2}*y*^{4}

(ii) 5*x* -10, 5*x*^{2}
– 20

(iii)*
x* ^{4} -1, *x*
^{2} − 2*x* + 1

(iv)
*x* ^{3} - 27, (*x*
- 3)^{2}, *x* ^{2} – 9

(i) 8*x** *^{4}*y*^{2},** **48

First let us find the
LCM of the numerical coefficients.

That is,
LCM (8, 48) = 2 × 2 × 2 ×6 = 48

Then find the LCM of the
terms involving variables.

That is, LCM (*x* ^{4}*y*^{2},
*x* ^{2}*y*^{4} ) = *x* ^{4}*y*^{4}

Finally find the LCM of
the given expression.

We condclude that the
LCM of the given expression is the product of the LCM of the numerical
coefficient and the LCM of the terms with variables.

Therefore, LCM (8*x*
^{4}*y*^{2}, 48*x* ^{2}*y*^{4} )
= 48*x* ^{4}*y*^{4}

(ii) (5*x* -10), (5*x*
^{2} - 20)

5*x* -10 = 5(*x*
− 2)

5*x* ^{2} -
20 = 5(*x* ^{2} − 4) = 5(*x* + 2)(*x* − 2)

Therefore, LCM [(5*x*
− 10),(5*x* ^{2} −20)] = 5(*x* + 2)(*x* −2)

(iii) (*x* ^{4}
-1), *x* ^{2} − 2*x* + 1

*x*^{4} -1 = (*x* ^{2} )^{2} −1 = (*x* ^{2}
+ 1)(*x* ^{2} −1) = (*x* ^{2} + 1)(*x* +
1)(*x* −1)

*x *^{2}* *−* *2*x *+* *1
= (*x* −1)^{2}

Therefore, LCM [(*x*
^{4} − 1),(*x* ^{2} −2*x* +1)] = (*x* ^{2}
+ 1)(*x* + 1)(*x* −1)^{2}

(iv)
*x *^{3}* *-* *27,* *(*x
*-* *3)^{2},* x *^{2}* *–* *9

*x *^{3}* *-* *27* *=* *(*x
*−* *3)(*x *^{2}* *+* *3*x *+* *9)*
*; (*x* − 3)^{2} = (*x* − 3)^{2} ; (*x*
^{2} − 9) = (*x* + 3)(*x* − 3)

Therefore, LCM [(*x*
^{3} - 27),(*x* - 3)^{2},(*x* ^{2} - 9)] = (*x*
− 3)^{2} (*x* + 3)(*x* ^{2} + 3*x* + 9)

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