14.2 Rational Expressions

14.2 Rational Expressions

Quick Answer: A rational expression is a fraction whose numerator and denominator are polynomials, such as $\frac{x^2 - 9}{x^2 - x - 6}$. Simplify by factoring completely, then cancel only common factors. Multiply by factoring all numerators and denominators and canceling across; divide by multiplying by the reciprocal. Always list excluded values—any $x$ that makes a denominator zero—from the original expression before simplifying.

Rational expressions show up on PERT whenever algebra moves past plain polynomials into fractions of expressions. The arithmetic is the same as numerical fractions, but the algebra requires careful factoring and a strict rule: only factors cancel.

What Is a Rational Expression?

Domain note: On PERT you usually do not write formal domain sets, but multiple-choice items frequently ask for values that make the expression undefined or for the simplified form for all allowed $x$.

Excluded Values (Critical First Step)

Excluded values (also called restrictions) are values of the variable that make any denominator equal to zero in the original expression—before you cancel factors.

Worked Example 1: Find excluded values

Problem: For $\frac{x^2 - 9}{x^2 - x - 6}$, which values of $x$ are excluded?

Step 1 — Factor the denominator only (for restrictions).

$$x^2 - x - 6 = (x - 3)(x + 2)$$

Step 2 — Set each factor to zero. $x - 3 = 0 \Rightarrow x = 3$; $x + 2 = 0 \Rightarrow x = -2$.

Step 3 — State exclusions. $x \neq 3$ and $x \neq -2$.

Even if a factor later cancels with the numerator, the value that made the original denominator zero remains excluded from the domain of the original expression. After canceling, you get an equivalent simplified expression on the common domain.

Simplifying: Factor, Then Cancel Factors Only

The golden rule

You may cancel a factor that multiplies the entire numerator with the same factor that multiplies the entire denominator. You may not cancel a term that is only part of a sum or difference.

Worked Example 2: Full simplify

Problem: Simplify $\frac{x^2 - 9}{x^2 - x - 6}$.

Step 1 — Factor numerator and denominator completely.

$$\text{Numerator: } x^2 - 9 = (x - 3)(x + 3)$$ $$\text{Denominator: } x^2 - x - 6 = (x - 3)(x + 2)$$

Step 2 — Cancel the common factor $(x - 3)$, noting $x \neq 3$ and also $x \neq -2$ from the other denominator factor.

$$\frac{(x - 3)(x + 3)}{(x - 3)(x + 2)} = \frac{x + 3}{x + 2}, \quad x \neq 3,\; x \neq -2$$

Step 3 — Check with a number. Let $x = 4$: original $\frac{16 - 9}{16 - 4 - 6} = \frac{7}{6}$; simplified $\frac{4 + 3}{4 + 2} = \frac{7}{6}$. Match.

Worked Example 3: Common factor with coefficients

Problem: Simplify $\frac{6x^2 + 3x}{9x}$.

Factor: $\frac{3x(2x + 1)}{9x} = \frac{3x(2x + 1)}{3 \cdot 3 \cdot x} = \frac{2x + 1}{3}$ for $x \neq 0$.

Trap: Writing $\frac{6x^2 + 3x}{9x} = 6x^2 + \frac{1}{3}$ by “splitting” incorrectly. Always factor first; do not cancel the $x$ in only one term of the numerator.

Multiplying Rational Expressions

Process:

1. Write the product as one big fraction: $\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}$.
2. Factor every polynomial in the numerator and denominator.
3. Cancel common factors across top and bottom.
4. Multiply remaining factors; state exclusions from all original denominators (and any denominator that appeared after a reciprocal in a division problem).

Worked Example 4: Multiply

Problem: Simplify $\frac{x^2 - 4}{x^2 + 5x + 6} \cdot \frac{x + 3}{x - 2}$.

Step 1 — Factor.

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

Step 2 — Cancel $(x - 2)$, $(x + 2)$, and $(x + 3)$ where allowed.

$$= 1, \quad \text{for } x \neq -2,\; x \neq -3,\; x \neq 2$$

(Every factor canceled; the product simplifies to the constant 1 on the common domain.)

Worked Example 5: Multiply with leftover factors

Problem: $\frac{2x}{x - 1} \cdot \frac{x^2 - 1}{4x}$.

$$\frac{2x}{x - 1} \cdot \frac{(x - 1)(x + 1)}{4x} = \frac{2x \cdot (x - 1) \cdot (x + 1)}{(x - 1) \cdot 4x} = \frac{x + 1}{2}, \quad x \neq 0,\; x \neq 1$$

Dividing Rational Expressions

Division flips the second fraction:

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

Then proceed exactly as in multiplication. Exclusions include zeros of $B$, zeros of $C$ (the reciprocal’s denominator), and zeros of $D$ if it was originally a denominator in a more nested form—on standard PERT items, track zeros of every denominator that appears before canceling.

Worked Example 6: Divide

Problem: Simplify $\frac{x^2 - 9}{x + 1} \div \frac{x - 3}{x^2 - 1}$.

Step 1 — Reciprocal multiply.

$$\frac{x^2 - 9}{x + 1} \cdot \frac{x^2 - 1}{x - 3}$$

Step 2 — Factor.

$$\frac{(x - 3)(x + 3)}{x + 1} \cdot \frac{(x - 1)(x + 1)}{x - 3}$$

Step 3 — Cancel $(x - 3)$ and $(x + 1)$.

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

Exclusions (list before canceling): The first denominator requires $x \neq -1$. The second expression $\frac{x - 3}{x^2 - 1}$ must be defined, so $x^2 - 1 \neq 0$ ⇒ $x \neq \pm 1$, and it must be nonzero for division, so $x \neq 3$. Combined: $x \neq -1$, $x \neq 1$, and $x \neq 3$.

Check with $x = 2$: Original $\frac{4 - 9}{2 + 1} \div \frac{2 - 3}{4 - 1} = \frac{-5}{3} \div \frac{-1}{3} = \frac{-5}{3} \cdot (-3) = 5$. Simplified $2^2 + 2(2) - 3 = 5$. Match.

On multiple choice, match the simplified polynomial (and the stated restrictions if both appear).

Common Errors Catalog

Worked Example 7: Spot the trap

Problem: Which is equivalent to $\frac{x^2 + 5x}{x}$ for $x \neq 0$?

A. $x + 5$
B. $x + 5x$
C. $5$
D. $x^2 + 5$

Correct: Factor $\frac{x(x + 5)}{x} = x + 5$ for $x \neq 0$. Choice A.
Why C is tempting: Students cancel the $x$ in $5x$ only and drop $x^2$, which is illegal.

Mini Decision Checklist

1. Identify every denominator in the original problem → excluded values.
2. Factor all numerators and denominators completely.
3. Cancel only identical factors (not terms).
4. Multiply or take reciprocal for products/quotients, then cancel again.
5. Write the simplified expression; attach exclusions if the question asks.
6. Spot-check with a safe number not equal to an excluded value.

PERT-Style Comparison Table

Rational expression items reward methodical factoring more than speed tricks. If you can factor linear and simple quadratic polynomials confidently, these problems become mechanical—and the main way to lose points is canceling something that is not a factor.