Least common multiple (L.C.M)

L.C.M by Factorization Method:

Finding the L.C.M of algebraic expressions by factorization involves factoring each expression into its irreducible factors, and then identifying the least common multiples.

Mathematically,

$\text{L.C.M} = \text{common factors}$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\text{non common factors}$

Here's an example to illustrate the process:

Example: Find the L.C.M of $x^3 - 8$ and $x^2+x-6$.

Solution:

Step 1: Factorize each expression.

$x^3 - 8^3 = x^3 - 2^3 = (x-2)(x^2+2x+4)$ $x^2 + x - 6 = (x+3)(x-2)$

Step 2: Figure out the common and non-common factors.

Common factors: $(x-2)$ Non Common Factors: $(x+3)(x^2+2x+4)$

Step 3: Multiply the common and non-common factors.

L.C.M $=$ Common Factors $\times$ Non Common Factors $= (x-2)\times(x+3)(x^2+2x+4)$ $= (x+3)(x^3-8)$

Hence the L.C.M of $x^3 - 8$ and $x^2+x-6$ is $(x+3)(x^3-8)$.