Least Common Multiple Of Algebraic Expressions

Least Common Multiple (LCM) with Polynomials

Example 1: Find the LCM of x3 + 5x2 + 6x and 2x3 + 4x2 Factor each expression (See Factoring Section): x3 + 5x2 + 6x → x(x + 2)(x + 3) 2x3 + 4x2 → 2x2 (x + 2)

2

x + 2

x

x(x+2)(x+3)

2x (x + 2)

x(x + 3)

2x

x + 3

2x

2

∴ LCM(x3 + 5x2 + 6x, 2x3 + 4x2 ) = x(x + 2)(x + 3)(2x)

Example 2: Find the LCM of x2 − 3x + 2, x2 − 5x + 6, and x2 − 7x + 12 Factor each expression (See Factoring Section): x2 − 3x + 2 → (x − 2)(x − 1) x2 − 5x + 6 → (x − 3)(x − 2) x2 − 7x + 12 → (x − 3)(x − 4)

x-2

(x-2)(x-1)

(x-3)(x-2)

(x-3)(x-4)

x-3

x-1

x-3

(x-3)(x-4)

x-1

1

x-4

∴ LCM(x2 − 3x + 2, x2 − 5x + 6, x2 − 7x + 12) = (x − 2)(x − 3)(x − 1)(x − 4)

1