1Cademy - Finding the LCD of $$\frac{8}{x^2-2x-3}$$ and $$\frac{3x}{x^2+4x+3}$$

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Rational Expression

Example

Finding the LCD of $\frac{8}{x^2-2x-3}$ and $\frac{3x}{x^2+4x+3}$

Find the least common denominator of the two rational expressions $\frac{8}{x^2 - 2x - 3}$ and $\frac{3x}{x^2 + 4x + 3}$ .

Step 1 — Factor each denominator completely.

$x^2 - 2x - 3 = (x + 1)(x - 3)$

$x^2 + 4x + 3 = (x + 1)(x + 3)$

Step 2 — List the factors and match common factors vertically. Both factored forms share the factor $(x + 1)$ . Align them so that matching factors appear in the same column:

First denominator: $(x + 1)$ and $(x - 3)$
Second denominator: $(x + 1)$ and $(x + 3)$

Step 3 — Bring down one factor from each column. The shared factor $(x + 1)$ is brought down once, along with the unshared factors $(x - 3)$ and $(x + 3)$ .

Step 4 — Multiply the factors. The LCD is $(x + 1)(x - 3)(x + 3)$ .

This example demonstrates how the column-matching technique extends from prime numbers to polynomial factors. Factoring both denominators reveals the common binomial factor $(x + 1)$ , which appears in both but is included only once in the LCD. The remaining unique factors $(x - 3)$ and $(x + 3)$ are each included as well, producing a three-factor LCD.

Updated 2026-04-30

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