Lesson 14: Rational Expressions

A rational expression is a fraction whose numerator and denominator are polynomials, like (x² − 9)/(x + 3). All the arithmetic of ordinary fractions applies — with one extra responsibility: values that make a denominator zero are excluded from the domain, permanently, even if they cancel away during simplifying. Division by zero is undefined, and algebra doesn't forgive it after the fact.

Simplifying: factor, then cancel factors

The golden rule: you may cancel common factors (things multiplied together), never common terms (things added). Factor everything first:

• (x² − 9)/(x + 3): numerator is a difference of squares, (x − 3)(x + 3); cancelling the factor (x + 3) leaves x − 3, valid for x ≠ −3.
• 6x²/(9x): cancel the common factor 3x → 2x/3, for x ≠ 0.
• (x² − 4)/(x² + 2x) = ((x − 2)(x + 2))/(x(x + 2)) = (x − 2)/x, for x ≠ 0 and x ≠ −2. Note two exclusions survive: both values that zeroed the *original* denominator.

State exclusions *before* simplifying — write them down from the original denominators, because after cancellation they're invisible and easy to lose.

Multiplying and dividing

Multiply straight across (factor first, cancel early): (2x/(x − 1)) · ((x − 1)/4) = x/2, the (x − 1) factors cancelling. Division means multiply by the reciprocal of the divisor — flip the *second* fraction only: (x²/3) ÷ (x/6) = (x²/3) · (6/x) = 2x. Verify with numbers at x = 2: original (4/3) ÷ (2/6) = (4/3) ÷ (1/3) = 4, and 2x = 4 ✓.

Adding and subtracting

With a common denominator, combine numerators: 3/x + 2/x = 5/x, and 5/(x − 2) − 3/(x − 2) = 2/(x − 2). With different denominators, build the least common denominator. For 1/(x + 1) + 2/x the LCD is x(x + 1). Rewrite each fraction: x/(x(x + 1)) + 2(x + 1)/(x(x + 1)), then combine: (x + 2x + 2)/(x(x + 1)) = (3x + 2)/(x(x + 1)). The numerator 3x + 2 shares no factor with the denominator, so this is fully simplified. Number-check at x = 1: original 1/2 + 2/1 = 2.5; simplified form (3 + 2)/(1 · 2) = 2.5 ✓ — substituting a test value is a fast, reliable audit of any rational-expression manipulation.

Modelling application: length from area

A rectangle's area is given by the polynomial x² + 5x + 6 (square units) and its width is x + 2. The length is area ÷ width: (x² + 5x + 6)/(x + 2) = ((x + 2)(x + 3))/(x + 2) = x + 3. At x = 4 the rectangle is 7 by 6, area 42 — and the polynomial gives 16 + 20 + 6 = 42 ✓. The exclusion x ≠ −2 is not pedantry here: at x = −2 the "width" would be 0 and the area formula 4 − 10 + 6 = 0 too — the model collapses exactly where the algebra forbids it. This factor-and-cancel pattern (a total divided into a rate, cost per student = (240 + 12n)/n, and so on) is how rational expressions appear whenever a whole is shared out.

Common mistakes

• Cancelling terms instead of factors. In (x + 3)/(x + 7) the x's are *terms of sums*, and nothing cancels — the expression is already simple. Cancellation only removes a factor shared by the *entire* numerator and the *entire* denominator. The test: can you factor it out of everything? (x − 3)(x + 3)/(x + 3) passes; (x + 3)/(x + 7) fails.
• Adding denominators. 1/2 + 1/3 is 5/6, not 2/5. Denominators are never added: build the LCD (6), convert (3/6 + 2/6), then add numerators. The polynomial version obeys the same law.
• Flipping the wrong fraction when dividing. (x²/3) ÷ (x/6): flip the divisor (x/6 → 6/x). Flipping the first fraction instead inverts the whole answer.
• Dropping the exclusions. (x² − 9)/(x + 3) simplifies to x − 3, but the original is undefined at x = −3 while x − 3 happily equals −6 there. Simplified forms inherit the original's restrictions — in exam language, the point is a *hole*, not a value.

Exam Experience

Question 1. For the expression 5/(x − 3), the excluded value is: A. x = 0 B. x = 5 C. x = 3 D. x = −3 Answer: C. The denominator x − 3 is zero exactly at x = 3. (A would apply to 5/x; D flips the sign.)

Question 2. Simplify (x² − 16)/(x + 4), and state the restriction. A. x − 4, x ≠ −4 B. x + 4, x ≠ 4 C. x − 4, no restriction D. x² − 4, x ≠ −4 Answer: A. (x² − 16) = (x − 4)(x + 4); cancel (x + 4) to get x − 4, but x = −4 still zeroes the *original* denominator, so the restriction survives. Option C is the answer of a student who cancelled and forgot.

Question 3. 2/a + 3/a equals: A. 5/(2a) B. 6/a C. 5/a² D. 5/a Answer: D. Common denominator a: numerators add, 2 + 3 = 5, denominator stays a. Option B multiplies the numerators (a multiplication reflex applied to addition); C adds denominators in disguise.

Question 4 (constructed response). A rectangle has area x² + 5x + 6 square units and width x + 2 units. (a) Find a simplified expression for its length, showing the factoring step. (b) State the excluded value of x and explain what happens to the rectangle there. (c) Find the length, width and area when x = 4, verifying the area in two ways. (d) Write an expression for the perimeter in terms of x and evaluate it at x = 4.

Scoring guide (4 points).

• 1 point: length = (x² + 5x + 6)/(x + 2) = ((x + 2)(x + 3))/(x + 2) = x + 3.
• 1 point: x ≠ −2, because the width x + 2 would be 0 — no rectangle exists (the area formula also gives 0 there).
• 1 point: at x = 4: width 6, length 7, area 42 — verified both by 6 × 7 and by the polynomial 4² + 5(4) + 6 = 42 ✓.
• 1 point: perimeter = 2((x + 3) + (x + 2)) = 4x + 10, which at x = 4 is 26 units (consistent with 2(7 + 6) = 26).

Deeper still: the average-speed trap and complex fractions

Average speed is not the average of speeds. Drive 60 km to a city at 30 km/h and return the same 60 km at 60 km/h. The naive average is 45 km/h — and it's wrong. Time out: 60/30 = 2 hours. Time back: 60/60 = 1 hour. Total: 120 km in 3 hours = 40 km/h. Speeds weight by *time spent*, and more time is spent at the slow speed. As a rational expression, the answer is 2d/(d/30 + d/60) = 2/(1/30 + 1/60) = 2/(3/60) = 40: the d cancels — the result doesn't even depend on the distance. This "harmonic" pattern governs every average-of-rates problem: fuel economy over equal distances, average pace over equal legs, combined machine rates. Complex fractions — fractions inside fractions — simplify by the same discipline: rewrite the overall division. (1 + 1/x)/(1 − 1/x): numerator is (x + 1)/x, denominator is (x − 1)/x, so the quotient is ((x + 1)/x) ÷ ((x − 1)/x) = (x + 1)/(x − 1), for x ≠ 0, 1. Audit at x = 2: original (1.5)/(0.5) = 3; simplified (3)/(1) = 3 ✓. Whether the expression is a speed average, an electrical parallel-resistance total, or an exam problem, the moves never change: combine within each level, flip the divisor, cancel factors, state exclusions.

One more idea: subtraction with an LCD, and dividing polynomials

Subtraction is where LCD work goes wrong, because the minus sign must distribute over the *whole* second numerator. Compute 5/x − 2/(x + 1): the LCD is x(x + 1); rewrite as 5(x + 1)/(x(x + 1)) − 2x/(x(x + 1)) = (3x + 5)/(x(x + 1)). The common error — writing the numerator as 5(x + 1) − 2 = 5x + 3 — misplaces the 2x. Audit at x = 1: original 5 − 2/2 = 4; formula (3 + 5)/(1 × 2) = 4 ✓. The numeric audit catches sign slips that staring at symbols never will. Division of polynomials inverts this lesson's modelling example: (2x² + 7x + 3) ÷ (x + 3) asks "what times (x + 3) gives the dividend?" — and Lesson 9 already factored the dividend as (2x + 1)(x + 3), so the quotient is 2x + 1. Polynomial division, at this level, *is* factoring viewed from the other side: every division problem is a factoring problem that has been kind enough to hand you one of its factors.

One more addition to lock the pattern in: 1/x + 2/(x + 1) has LCD x(x + 1), giving ((x + 1) + 2x)/(x(x + 1)) = (3x + 1)/(x(x + 1)). Audit at x = 1: the original is 1 + 2/2 = 2, and the combined form is (3 + 1)/(1 × 2) = 2 ✓. Note the exclusions x ≠ 0, −1 ride along silently — the simplified form is defined at neither, exactly as the original demands. Build the reflex: LCD, rewrite, combine, factor-check the numerator, audit with a number, state the exclusions. Six moves, every problem.

Rational expressions also run the world's unit conversions, in disguise as unit-factor chains: 60 miles per hour becomes feet per second by multiplying by fractions equal to 1 — (5280 ft/1 mi) and (1 h/3600 s) — chosen so the unwanted units cancel like algebraic factors: 60 × 5280/3600 = 88 ft/s. The cancellation is literal: miles cancel miles, hours cancel hours, exactly as (x + 3) cancelled earlier. Every conversion in science — km/h to m/s, dollars per dozen to cents each — is this same rational-expression arithmetic with units riding along as factors, and a conversion whose units fail to cancel is wrong before its arithmetic is even checked.