Operations with Rational Expressions

➗ Operations with Rational Expressions

Part 1 of 5 — Simplifying & Domain Restrictions

Topics in This Part

🔑 Key Concept: A rational expression is just a fraction whose numerator and denominator are polynomials. Every operation you already know for number fractions — simplify, multiply, divide, add, subtract — carries over. The new twist is that we must factor first and always watch the denominator.

What Is a Rational Expression?

A rational expression is a quotient of two polynomials:

$\frac{P(x)}{Q(x)}, \quad \text{where } Q(x) \ne 0$

⚠️ The denominator can never equal zero. Division by zero is undefined, so any $x$-value that makes $Q(x) = 0$ is excluded from the domain. Finding those values is step zero of every problem.

Finding Excluded Values

To find the excluded values, set the original denominator equal to zero and solve.

Example: $\dfrac{x+4}{x^2 - 9}$

Factor the denominator: $x^2 - 9 = (x-3)(x+3)$.

Set each factor to zero: $x - 3 = 0 \;\Rightarrow\; x = 3, \qquad x + 3 = 0 \;\Rightarrow\; x = -3$

So $x \ne 3$ and $x \ne -3$. The domain is "all real numbers except $3$ and $-3$."

💡 Use the ORIGINAL denominator. Always hunt for excluded values before you cancel anything — a factor you cancel away still restricts the domain (more on that in a moment).

Concept Check 🎯

A Quick Routine

For now, our denominators are simple. The routine never changes:

1. Look at the denominator only.
2. Set it equal to $0$.
3. Solve. Each solution is an excluded value.

For a linear denominator like $ax + b$, there's exactly one excluded value: $x = -\dfrac{b}{a}$.

Find the Excluded Value 🧮

Enter the single $x$-value that must be excluded.

1) $\dfrac{7}{x - 6}$,   excluded $x = \,?$ 2) $\dfrac{x}{x + 2}$,   excluded $x = \,?$ 3) $\dfrac{x-1}{2x - 10}$,   excluded $x = \,?$

Simplifying: Factor, Then Cancel

To simplify a rational expression you factor the numerator and denominator completely, then cancel any factor that appears in both.

🔑 Golden Rule: You may cancel factors (things multiplied), never terms (things added or subtracted). $\dfrac{x+3}{x}$ does NOT simplify to $3$.

Example: $\dfrac{x^2 - 9}{x^2 + 7x + 12}$

Factor both: $\frac{x^2 - 9}{x^2 + 7x + 12} = \frac{(x-3)(x+3)}{(x+3)(x+4)}$

Cancel the common factor $(x+3)$: $= \frac{x-3}{x+4}$

⚠️ The cancelled factor still restricts the domain. Here $(x+3)$ was cancelled, but $x = -3$ is still excluded — the simplified form $\dfrac{x-3}{x+4}$ carries hidden restrictions $x \ne -4$ and $x \ne -3$.

Simplify Step by Step 🔽

Simplify $\dfrac{x^2 - 25}{x^2 - 2x - 15}$.

➗ Operations with Rational Expressions

Part 2 of 5 — Multiplying & Dividing

🔑 The Idea: Multiplying and dividing rational expressions works exactly like multiplying and dividing number fractions — factor everything, then cancel across the whole product before you multiply out.

Multiplying Rational Expressions

$\frac{A}{B} \cdot \frac{C}{D} = \frac{A \cdot C}{B \cdot D}$

The smart order: factor → cancel any factor on top with any factor on the bottom → multiply what's left.

Example: $\dfrac{x+2}{x-3} \cdot \dfrac{x^2 - 9}{x^2 + 5x + 6}$

Factor everything: $\frac{x+2}{x-3} \cdot \frac{(x-3)(x+3)}{(x+2)(x+3)}$

Cancel $(x+2)$, $(x-3)$, and $(x+3)$ — all appear on both top and bottom: $= \frac{\cancel{(x+2)}}{\cancel{(x-3)}} \cdot \frac{\cancel{(x-3)}\,\cancel{(x+3)}}{\cancel{(x+2)}\,\cancel{(x+3)}} = 1$

💡 Because every factor cancelled, the product simplifies to $1$ (for allowed $x$). Cancelling before multiplying keeps the numbers small.

Concept Check 🎯

Dividing Rational Expressions

To divide, multiply by the reciprocal of the second expression ("keep–change–flip"):

$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$

Example: $\dfrac{x^2 - 4}{x + 1} \div \dfrac{x + 2}{x^2 - 1}$

Flip the second fraction and multiply: $\frac{x^2 - 4}{x + 1} \cdot \frac{x^2 - 1}{x + 2}$

Factor: $x^2 - 4 = (x-2)(x+2)$ and $x^2 - 1 = (x-1)(x+1)$. $\frac{(x-2)(x+2)}{x+1} \cdot \frac{(x-1)(x+1)}{x+2}$

Cancel $(x+2)$ and $(x+1)$: $= (x-2)(x-1)$

⚠️ Only flip the divisor (the fraction after the $\div$). Flipping the wrong one is the #1 division error.

Divide Step by Step 🔽

Simplify $\dfrac{x^2 - 16}{x} \div \dfrac{x + 4}{x^2}$.

Watch the Factors Vanish

When a factor appears identically on top and bottom — even a whole binomial like $(x-7)$ — it cancels to $1$. After everything cancels, you're often left with a plain number.

💡 In the drill below, each product is built so that all the $x$-factors cancel. Track them carefully and a single number should remain.

Multiply & Divide 🧮

Each answer below simplifies to a single number. Compute it.

1) $\dfrac{6}{x} \cdot \dfrac{x}{3} = \,?$ 2) $\dfrac{x^2}{5} \div \dfrac{x^2}{10} = \,?$ 3) $\dfrac{x-7}{4} \cdot \dfrac{12}{x-7} = \,?$

➗ Operations with Rational Expressions

Part 3 of 5 — Adding & Subtracting (Like Denominators) and the LCD

🔑 The Idea: To add or subtract fractions you need a common denominator. With like denominators it's instant; with unlike denominators you first build the Least Common Denominator (LCD) out of the factors.

Like Denominators

When the denominators are the same, add or subtract the numerators and keep the denominator:

$\frac{A}{D} \pm \frac{B}{D} = \frac{A \pm B}{D}$

Example: $\dfrac{2x}{x+3} + \dfrac{6}{x+3}$

$= \frac{2x + 6}{x+3} = \frac{2(x+3)}{x+3} = 2$

Example (subtraction): $\dfrac{5x}{x-1} - \dfrac{5}{x-1}$

$= \frac{5x - 5}{x-1} = \frac{5(x-1)}{x-1} = 5$

⚠️ Subtraction traps everyone. The minus sign applies to the entire numerator. Wrap it in parentheses: $\dfrac{A}{D} - \dfrac{B}{D} = \dfrac{A - (B)}{D}$, then distribute.

Concept Check 🎯

Building the LCD

When denominators differ, the Least Common Denominator is the product of each distinct factor raised to its highest power across all denominators.

Example: denominators $6x^2$ and $9x$

• $6x^2 = 2 \cdot 3 \cdot x^2$
• $9x = 3^2 \cdot x$

Take the highest power of each: $2^1$, $3^2$, $x^2$. $\text{LCD} = 2 \cdot 9 \cdot x^2 = 18x^2$

Example: denominators $(x+1)$ and $(x+1)(x-2)$

The second already contains the first, so: $\text{LCD} = (x+1)(x-2)$

💡 Factor first, always. You can't see shared factors like $(x+1)$ until each denominator is fully factored.

Find the LCD 🔽

Don't Stop at "Combine"

After combining numerators over a like denominator, always try to factor the new numerator — it may cancel with the denominator.

Example: $\dfrac{3x}{x+4} + \dfrac{12}{x+4}$

$= \frac{3x + 12}{x+4} = \frac{3(x+4)}{x+4} = 3$

🔑 Factoring the numerator is what turns a messy-looking sum into a clean answer.

Combine Like Denominators 🧮

Each expression simplifies to a single number. Compute it.

1) $\dfrac{3x}{x+4} + \dfrac{12}{x+4} = \,?$ 2) $\dfrac{7x}{x-2} - \dfrac{14}{x-2} = \,?$

➗ Operations with Rational Expressions

Part 4 of 5 — Unlike Denominators & Complex Fractions

🔑 Big Payoff: With the LCD in hand, you rewrite each fraction so they share it, then add or subtract the numerators. This is the most common — and most tested — rational-expression skill.

Adding with Unlike Denominators

The 4 steps:

1. Factor all denominators and find the LCD.
2. Multiply each fraction by the factor it's missing (top and bottom).
3. Combine the numerators over the common LCD.
4. Simplify the result.

Example: $\dfrac{2}{x} + \dfrac{3}{x+1}$

The LCD is $x(x+1)$. Rewrite each fraction: $\frac{2}{x}\cdot\frac{x+1}{x+1} + \frac{3}{x+1}\cdot\frac{x}{x} = \frac{2(x+1)}{x(x+1)} + \frac{3x}{x(x+1)}$

Combine numerators: $= \frac{2x + 2 + 3x}{x(x+1)} = \frac{5x + 2}{x(x+1)}$

💡 The numerator $5x+2$ doesn't factor, so this is fully simplified.

Concept Check 🎯

A Subtraction Example

Example: $\dfrac{x}{x-2} - \dfrac{4}{x+2}$

The denominators $x-2$ and $x+2$ share no factors, so the LCD is $(x-2)(x+2)$.

$\frac{x(x+2)}{(x-2)(x+2)} - \frac{4(x-2)}{(x-2)(x+2)}$

Now subtract the whole second numerator (parentheses!): $= \frac{x^2 + 2x - (4x - 8)}{(x-2)(x+2)} = \frac{x^2 + 2x - 4x + 8}{(x-2)(x+2)}$

$= \frac{x^2 - 2x + 8}{(x-2)(x+2)}$

⚠️ Notice $-(4x - 8)$ became $-4x + 8$. Forgetting to flip that $-8$ to $+8$ is the most common subtraction mistake.

Complex Fractions

A complex fraction has a fraction inside a fraction. The cleanest method: multiply the top and bottom by the LCD of all the little denominators.

Example: $\dfrac{\;\dfrac{1}{x} + 1\;}{\;\dfrac{1}{x} - 1\;}$

The only inner denominator is $x$, so multiply top and bottom by $x$: $\frac{x\left(\dfrac{1}{x} + 1\right)}{x\left(\dfrac{1}{x} - 1\right)} = \frac{1 + x}{1 - x}$

💡 Multiplying through by the LCD clears all the small fractions in one move — far cleaner than dividing fraction-by-fraction.

Build the Sum 🔽

You are adding $\dfrac{3}{x-1} + \dfrac{2}{x}$. Choose each step.

Same Rules with Plain Numbers

These ideas aren't new — they're the fraction rules you've always used, now stated carefully. Adding $\dfrac{1}{3} + \dfrac{1}{x}$ uses the LCD $3x$ exactly like $\dfrac{1}{3} + \dfrac{1}{4}$ uses the LCD $12$.

💡 If a step ever feels strange, test it on plain numbers first — the structure is identical.

Combine & Evaluate 🧮

1) Add $\dfrac{1}{3} + \dfrac{1}{x}$. The combined numerator is $x + a$. Enter $a$. 2) For $\dfrac{1}{3} + \dfrac{1}{x}$, the common denominator is $3x$. Enter the number multiplying $x$ in it (i.e. the coefficient):   $\,?$ 3) Simplify $\dfrac{\frac{1}{2}}{\frac{1}{4}}$ to a single number: $\,?$

➗ Operations with Rational Expressions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) simplify and find excluded values, (2) multiply and divide, (3) build the LCD, and (4) add and subtract with any denominators. Let's put it all together.

Quick Reference

⚠️ Two habits prevent most errors: factor before you do anything, and wrap a subtracted numerator in parentheses so the minus sign distributes.

Mixed Practice 🎯

Before the Final Check

One last warm-up that touches every skill: a multiply-and-cancel, a combine-and-factor, and an excluded-value hunt.

🔑 Remember the order of operations on a rational expression: factor first, then cancel, combine, or read off the restriction.

One More Set 🧮

Each answer is a single number. Compute it.

1) $\dfrac{x^2 - 16}{x+4} \cdot \dfrac{1}{x-4} = \,?$ 2) Simplify $\dfrac{5x + 5}{x + 1} = \,?$ 3) Excluded value of $\dfrac{x+9}{x - 8}$:   $x = \,?$

You're Ready

You've simplified, multiplied, divided, found LCDs, added, subtracted, and tamed complex fractions. Three questions stand between you and mastery of rational-expression operations.

💡 Take your time, factor first, and mind every minus sign.

Exit Quiz ✅

Answer all three to finish the lesson.