Adding & Subtracting Rational Expressions

Find common denominators and combine rational expressions. Math 20-1 Alberta curriculum.

Lesson 6.4 of Rational Expressions and Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Finding the Lowest Common Denominator (LCD). Factor each denominator, list every unique factor at its highest power, and multiply them together to find the LCD of rational expressions.
• Adding and Subtracting Rational Expressions. Find NPVs, build the LCD, rewrite each fraction, combine numerators, and simplify — applied to progressively complex rational expressions including trinomial denominators.
• Complex Fractions and Word Problems. Collapse nested fractions into a single fraction then divide, and translate real-world travel scenarios into rational expressions to model total time.

Inside the lesson — a free preview

What Is the LCD?

Defining the LCD. The lowest common denominator (LCD) of two or more fractions is the smallest expression that all the denominators divide into evenly. The same idea you used with regular fractions applies here — but now the denominators may be polynomials.

4 Steps to Find the LCD.

Key Tip — LCD vs. Product of Denominators. The LCD is not simply the product of all the denominators. Multiplying all denominators together always gives a common denominator, but usually not the lowest one. Using the true LCD keeps your numbers smaller and your final answer easier to simplify.

Quick LCD Practice

Each example below builds on the last. Work through them to see how the LCD grows as denominators become more complex.

a) LCD of (1)/(6) and (2)/(5). Factor each denominator into primes:

6 = 2 · 3

5 = 5

Unique factors: 2, 3, 5.

LCD = 2 · 3 · 5

LCD = 30

b) LCD of (1)/(6x) and (2)/(5x^2). Factor each denominator:

6x = 2 · 3 · x

5x^2 = 5 · x^2

Unique factors: 2, 3, 5, x. The highest power of x that appears is x^2.

LCD = 2 · 3 · 5 · x^2

LCD = 30x^2

c) LCD of (1)/(6xy) and (2)/(5x^2). Factor each denominator:

6xy = 2 · 3 · x · y

5x^2 = 5 · x^2

Unique factors: 2, 3, 5, x, y. Highest power of x is x^2.

LCD = 2 · 3 · 5 · x^2 · y

LCD = 30x^2y

d) LCD of (1)/(6(x+1)) and (2)/(5(x-3)). The variable factors here are (x+1) and (x-3) — two different binomials, so both must appear in the LCD.

LCD = 2 · 3 · 5 · (x+1) · (x-3)

LCD = 30(x+1)(x-3)

e) LCD of (1)/(6x(x+1)) and (2)/(5(x-3)). Now the first denominator also has an extra factor of x, plus the two distinct binomials.

LCD = 2 · 3 · 5 · x · (x+1) · (x-3)

LCD = 30x(x+1)(x-3)

Critical Rule — Different Binomials Are Always Separate. Different binomials such as (x+1) and (x-3) are not common factors. Each one must appear in the LCD on its own. Only repeated factors can be combined using a higher power.

Worked example: a — Monomial Denominators

Simplify (1)/(6xy) - (2)/(15x^2).

Example 1a — (1)/(6xy) - (2)/(15x^2). Step 1 — Find NPVs from the original denominators.

6xy = 0 when x = 0 or y = 0

15x^2 = 0 when x = 0

So x ≠ 0 and y ≠ 0.

Step 2 — Find the LCD.

6xy = 2 · 3 · x · y

15x^2 = 3 · 5 · x^2

Unique factors with highest powers: 2, 3, 5, x^2, y.

LCD = 30x^2y

Step 3 — Rewrite the first fraction with denominator 30x^2y.

Multiply 6xy by 5x to reach 30x^2y:

(1)/(6xy) = (1 · 5x)/(6xy · 5x) = (5x)/(30x^2y)

Step 4 — Rewrite the second fraction with denominator 30x^2y.

Multiply 15x^2 by 2y to reach 30x^2y:

(2)/(15x^2) = (2 · 2y)/(15x^2 · 2y) = (4y)/(30x^2y)

Step 5 — Subtract the numerators.

(5x)/(30x^2y) - (4y)/(30x^2y) = (5x - 4y)/(30x^2y)

Step 6 — Check for further simplification.

The numerator (5x - 4y) shares no factor with 30x^2y, so this is fully reduced.

Answer: (5x - 4y)/(30x^2y), where x ≠ 0 and y ≠ 0

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