Lowest Common Multiple (LCM) of Algebraic Expressions

The lowest common multiple (LCM) of two or more algebraic expressions is the simplest expression that is divisible by each individual expression. Finding the LCM is an essential step in simplifying rational expressions and solving algebraic equations.

The Step-by-Step Method

  1. Factor: Factor each algebraic expression completely.
  2. Identify Factors: List every unique factor present across all expressions.
  3. Highest Power: For factors common to multiple expressions, select the one with the highest exponent.
  4. Multiply: The product of these selected factors is the LCM.

Solved Examples

Example 1: Difference of Squares

Find the LCM of: \(x^2 - 1\) and \(x - 1\).

View Solution

1. Factor each expression:

\[ x^2 - 1 = (x - 1)(x + 1) \] \[ x - 1 = (x - 1) \]

2. Identify unique factors: \((x - 1)\) and \((x + 1)\).

3. Use highest powers: Both \((x-1)\) and \((x+1)\) appear with power 1.

LCM: \( \mathbf{(x - 1)(x + 1)} \)

Example 2: Monomials and Binomials

Find the LCM of: \(2x^2\), \(x^2 + x\), and \(x^3 + 2x\).

View Solution

1. Factor completely:

\[ 2x^2 = 2 \cdot x^2 \] \[ x^2 + x = x(x + 1) \] \[ x^3 + 2x = x(x^2 + 2) \]

2. Select factors with highest powers:

  • Numerical coefficient: \(2\)
  • Variable \(x\): \(x^2\) is the highest power.
  • Binomials: \((x + 1)\) and \((x^2 + 2)\).

LCM: \( \mathbf{2x^2 (x + 1) (x^2 + 2)} \)

Example 3: Trinomials

Find the LCM of: \(x^2 + 3x - 4\), \((x - 1)^2\), and \(x^2 + 9x + 20\).

View Solution

1. Factor each trinomial:

\[ x^2 + 3x - 4 = (x - 1)(x + 4) \] \[ (x - 1)^2 = (x - 1)^2 \] \[ x^2 + 9x + 20 = (x + 4)(x + 5) \]

2. Apply the highest power rule:

  • Factor \((x - 1)\): use \((x - 1)^2\).
  • Factor \((x + 4)\): use once.
  • Factor \((x + 5)\): use once.

LCM: \( \mathbf{(x - 1)^2 (x + 4)(x + 5)} \)

Practice Questions

1. Find LCM of: \( 2 (x + 1) \) and \( 3 (x + 1) \)

View Solution

Coefficients 2 and 3 have LCM 6. Common factor is \((x+1)\).
Result: \( \mathbf{6(x + 1)} \)

2. Find LCM of: \( 2 (x - 1)^2 \) and \( 5 (x - 1) \)

View Solution

Coefficients 2 and 5 have LCM 10. Highest power of \((x-1)\) is \((x-1)^2\).
Result: \( \mathbf{10(x - 1)^2} \)

3. Find LCM of: \( x^2 + 5x + 6 \) and \( 2x^2 + 2x - 4 \)

View Solution
\[ x^2 + 5x + 6 = (x + 3)(x + 2) \] \[ 2x^2 + 2x - 4 = 2(x^2 + x - 2) = 2(x + 2)(x - 1) \]

LCM: \( \mathbf{2(x + 3)(x + 2)(x - 1)} \)

4. Find LCM of: \( 3x^3 - 2x^2 - x \) and \( x - 1 \)

View Solution
\[ 3x^3 - 2x^2 - x = x(3x^2 - 2x - 1) = x(3x + 1)(x - 1) \]

The other expression is simply \((x-1)\).
LCM: \( \mathbf{x(3x + 1)(x - 1)} \)

5. Find LCM of: \( 3x^3 - 2x^2 - x \), \( 2x^2 - 2 \), and \( (x - 1)^2 \)

View Solution
\[ 3x^3 - 2x^2 - x = x(3x + 1)(x - 1) \] \[ 2x^2 - 2 = 2(x^2 - 1) = 2(x - 1)(x + 1) \] \[ (x - 1)^2 = (x - 1)^2 \]

Selecting highest powers: \(2, x, (3x+1), (x+1), (x-1)^2\).
LCM: \( \mathbf{2x(3x + 1)(x + 1)(x - 1)^2} \)

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