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Theory:

The Least Common Multiple of two or more algebraic expressions is the expression of the lowest degree (or power) which is divisible by each of them without remainder.
The LCM of a number or an algebraic expression by factorization method can be determined using the following steps:

(i) Each expression must be resolved into its simple factors.

(ii) The highest power of the common 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.
Example:
1. Find the LCM of \(16x^3y^2z, 45xy^9z^7,12x^9y^2z\)
 
Solution:
 
Step 1: Let us find the LCM of the numerical coefficients.
 
\(16 = 2^4\), \(45 = 3 \times 3 \times 5\), \(12 = 2 \times 2 \times 3\)
 
\(LCM(16, 45, 12) = 2^4 \times 3 \times 3 \times 5 = 720\)
 
Step 2: Let us find the LCM of the variable terms.
 
\(LCM(x^3y^2z, xy^9z^7, x^9y^2z) = x^9y^9z^7\)
 
Step 3: The product of the LCM of the numerical coefficients and variable terms is the required LCM.
 
\(LCM(16x^3y^2z, 45xy^9z^7,12x^9y^2z) = 720x^9y^9z^7\)
 
Thus, the required LCM is \(720x^9y^9z^7\).
 
 
2. Find the LCM of \(21(x^4 - x^2), 16(x^2 + 3x)^2\).
 
Solution:
 
\(21(x^4 - x^2) = 3 \times 7 \times x^2 \times (x^2 - 1)\)
 
\(16(x^2 + 3x)^2 = 2^4 \times (x^4 + 6x^3 + 9x^2) = 2^4 \times x^2 \times (x^2 + 6x + 9) = 2^4 \times x^2 \times (x + 3)(x + 3)\)
 
The LCM of \(21(x^4 - x^2), 16(x^2 + 3x)^2\) is given by:
 
\(LCM = 3 \times 7 \times 2^4 \times x^2 \times (x + 1)(x - 1) \times (x + 3)(x + 3)\)
 
\(= 336 \times x^2(x^2 -1)(x + 3)^2\)
 
Therefore, \(LCM(21(x^4 - x^2), 16(x^2 + 3x)^2) = 336 \times x^2(x^2 -1)(x + 3)^2\).