A fraction whose bottom hides a letter — and the new rule it forces on us.
Point 2 of 4 in this lesson: 9.1.2 The denominator test: rational or just a polynomial?
Suppose you and some friends order a $60 pizza and split the cost evenly. If three of you share, each pays $20. If four share, $15. Five share, $12. The number of friends keeps changing, so let’s give it a letter: call it x. Then the cost each of you pays is 60 ÷ x, which we write as the fraction 60x. That little fraction — a number on top, a letter on the bottom — is your first rational expression. And it comes with a brand-new rule.
By the end of this lesson you will be able to: (1) say exactly what a rational expression is, (2) tell a rational expression apart from an ordinary polynomial using the denominator test, (3) find the values of x a rational expression is not allowed to take — its excluded values — and (4) decide when a rational expression equals zero. Throughout Stage 9 we keep one color code: the numerator (top) is amber, the denominator (bottom) is blue, a cancelled factor is green, and a forbidden value is red.
Back in Stage 3 you met ordinary fractions like 34 — a number on top, a number on the bottom, and the bar means divide. A rational expression keeps that exact picture but lets letters move in. Formally:
A rational expression is anything of the form AB where A and B are polynomials and the bottom B contains a variable. The bar still means divide: AB means A ÷ B.
Here are a few, with the bottom painted blue so you can see the letter living downstairs:
That is the whole definition. A rational expression is just a polynomial divided by a polynomial, with the understanding that the bottom is carrying a letter. Notice that everything you already know about fractions — same top and bottom means 1, the bar means divide, you can reduce — is going to carry straight over. The catch (there is always a catch) waits for us in §9.1.3.
The cost-each fraction 60x is a rational expression: top is the polynomial 60, bottom is the polynomial x. Same shape as 603 = 20, but now the bottom can change.
Tap each expression to file it under Polynomial or Rational expression. The denominator lights up blue — if a variable lives in it, it’s rational. Watch the reason that appears.
Here is a subtlety worth slowing down for. Look at x3. There is a letter in it — but the letter is on top, and the bottom is just the number 3. Dividing by 3 is the same as multiplying by ⅓, so x3 = ⅓ x. That is an ordinary polynomial — no variable is ever in a denominator.
Look only at the bottom. If the denominator is just a number, the expression is a polynomial in disguise. Only when a variable sits in the denominator do you have a genuine rational expression. Bottom has a letter ⇒ rational.
Sort these six by that one test:
Every polynomial is technically a rational expression — you can always write 7 as 71 or x²−1 as x²−11. But the new objects this whole stage is about are the ones with a real variable downstairs. Those are the ones that can misbehave.
Step through the six expressions. For each, predict Polynomial or Rational, then reveal. The bottom is highlighted to remind you where to look.
Now the catch. Go back to the pizza: 60x is the cost when x people share. What happens if x = 0? You’d be splitting a pizza among nobody. There is no “cost each” — the question is meaningless. And in arithmetic, dividing by 0 is forbidden, full stop. So 60x is undefined at x = 0.
A rational expression is undefined wherever its denominator equals 0. To find these excluded values, set the denominator equal to 0 and solve (factor first if you must). Every rational expression travels with this short no-go list.
This is the new idea of Stage 9, so let’s do three of them carefully.
(a) x+1x−2: set x − 2 = 0 ⇒ x = 2. Just one forbidden point.
(b) 5x²−9: factor the bottom, x² − 9 = (x − 3)(x + 3). A product is 0 only when a factor is 0, so x = 3 or x = −3. Two forbidden points.
(c) 1x(x+1): the bottom is already factored, x(x + 1). It’s 0 when x = 0 or x = −1.
You can see the no-go list on a number line: punch an open hole at every forbidden value. The expression lives everywhere on the line except inside those holes.
What about xx²+1? Set x² + 1 = 0 ⇒ x² = −1. No real number squares to a negative, so the bottom is never 0. This expression has no excluded values at all — its no-go list is empty. Always check; don’t assume there must be a hole.
Pick a denominator. The widget factors it, lists the forbidden x-values, and pokes open holes in the number line. Then slide x toward a hole and watch the value of 1B balloon.
One more honest question about fractions. When is an ordinary fraction equal to 0? Think of 05 = 0, but 50 is undefined. A fraction is 0 exactly when its top is 0 — and, quietly, its bottom is not 0. The same is true for rational expressions.
A rational expression equals 0 only when its numerator is 0 and its denominator is not 0 there. Set the top = 0 and solve — then double-check each answer is allowed (not on the no-go list).
x−3x+1 = 0. Set the top x − 3 = 0 ⇒ x = 3. Is 3 allowed? The bottom is 3 + 1 = 4, which is not 0 — so yes. The expression really is 0 at x = 3. (And it is undefined at x = −1, where the bottom dies.)
But here is a trap that catches almost everyone the first time.
Take x−2x²−4. At x = 2 the top is 2 − 2 = 0 — so is it zero there? No. The bottom is 2² − 4 = 0 too. A value that zeroes both top and bottom doesn’t give 0 — it gives 0 ÷ 0, which is undefined. So x = 2 is on the no-go list, not a zero of the expression.
So the full verdict for any single x has three possible outcomes, and you check the bottom first:
| If the bottom is… | and the top is… | the value is… |
|---|---|---|
| 0 | anything | undefined |
| not 0 | 0 | 0 |
| not 0 | not 0 | a plain number |
Slide x through x−3x+1. The machine reports the top, the bottom, and the verdict. Find where it reads = 0 and where it reads undefined.
And to close the loop with the real world: the same three-part verdict shows up in every formula with a variable on the bottom. Time is distance ÷ speed, so a 120 km trip at speed v takes 120v hours — undefined at v = 0 (a car that never moves never arrives). A salt concentration is saltsalt + water — undefined when there’s nothing in the cup at all. Rational expressions are how we describe quantities that depend on a changing amount.
A rational expression is a fraction AB whose bottom carries a variable — the bar still means divide. The denominator test tells it apart from a polynomial: a letter must be downstairs. Because that bottom depends on x, it can secretly hit 0, and division by 0 is forbidden — so every rational expression carries a no-go list of excluded values, found by setting the bottom = 0 and solving. And it equals 0 only when the top is 0 while the bottom is alive — if both die at once, it’s undefined, not zero.
In 9.2 we’ll start simplifying these fractions — factoring top and bottom and cancelling common factors, exactly like reducing 69 to 23, but the no-go list you met today stays with us the whole way. Later come multiplying and dividing (9.3), adding and subtracting (9.4), and solving rational equations (9.5), where forgetting the no-go list invents fake answers.
Which of these are rational expressions (variable in the bottom)? x7, 7x, x²+14, 4x²+1.
Find the excluded value(s) of x+5x−7.
Find all excluded values of 3x²−25.
Find all excluded values of xx(x−4).
Does 2xx²+9 have any excluded values?
For what value of x is x+6x−1 equal to 0?
Is x+4x²−16 equal to 0 when x = −4?
Evaluate x−3x+1 at x = 5, and state its verdict at x = −1.
A $48 bill is split evenly among n people. Write the cost each as a rational expression and give its excluded value. What is the cost when n = 6?
A trip of 150 km is driven at speed v km/h. Write the time as a rational expression, state its excluded value, and find the time when v = 50.
Six questions to lock it in. Tap the answer you think is right.
This lesson extends students’ fraction sense (Stage 3) to expressions with a variable denominator, and foreshadows the operations on rational expressions that follow. It builds toward the US Common Core standards A-APR.D.6 and A-APR.D.7 (rewriting and operating on rational expressions, and closure under the field operations), draws on A-SSE.A.2 (seeing structure — factoring the denominator to find where it is 0), and connects back to 7.EE work with expressions. The central new concept — that a rational expression is undefined wherever its denominator is 0 — is the domain idea underlying all of A-APR and the gateway to A-REI.A.2 (extraneous roots) in 9.5.
The #1 misconception is treating a rational expression as “just a fraction” and forgetting the denominator-can’t-be-0 restriction — students happily plug in the very value that breaks the expression. The antidote built into this page: every single example states its excluded value(s) out loud, and the three-row verdict table forces the habit of checking the bottom first. Encourage your student to write the no-go list before doing anything else with a rational expression — it’s a discipline that pays off through 9.5.