A rational expression is a fraction whose denominator hides a letter. Because the bottom now depends on x, it can secretly become 0 — and that one point is forbidden.
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.
9.1.1 What is a rational expression?
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:
Key idea
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:
Three rational expressions. In each one a polynomial sits on top and a polynomial with a variable sits on the bottom.
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.
Where it comes from
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.
🎮 Try itRATIONAL-OR-NOT SORTER
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.
Tap an expression above to sort it.
9.1.2 The denominator test: rational or just a polynomial?
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.
The denominator test
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:
Same letters, opposite answers. What decides it is one thing only: is there a variable on the bottom?
A fine print worth knowing
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.
🎮 Try itDENOMINATOR DETECTIVE
Step through the six expressions. For each, predict Polynomial or Rational, then reveal. The bottom is highlighted to remind you where to look.
Your call:
Expression 1 of 6. Make your call.
9.1.3 When does it make sense? Excluded values
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.
Key idea — the excluded values
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.
Example — three no-go lists
(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.
The no-go list of 5x²−9: open holes at x = −3 and x = 3. Everywhere else, the expression is perfectly fine.
Watch — some bottoms never hit 0
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.
🎮 Try itEXCLUDED-VALUE FINDER
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.
Denominator B =
x
9.1.4 When does it equal zero?
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.
Key idea — the zero test
A rational expression equals 0 only when its numerator is 0and 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).
Example — a clean zero
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.
Watch — the “both are zero” trap
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
Always look at the bottom first. If it’s 0, you stop — undefined. Only with a living bottom does a 0 on top make the whole thing 0.
🎮 Try itZERO / UNDEFINED / VALUE MACHINE
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.
x
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.
★ The big ideas, in one breath
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 0while the bottom is alive — if both die at once, it’s undefined, not zero.
What’s next
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.
✎ Exercises 9.1
Which of these are rational expressions (variable in the bottom)? x7, 7x, x²+14, 4x²+1.
Show answer
Rational: 7x and 4x²+1 (a variable sits in the bottom). The other two have number-only bottoms, so they are polynomials.
Find the excluded value(s) of x+5x−7.
Show answer
Set the bottom x − 7 = 0 ⇒ x = 7. The top is ignored when finding excluded values.
Find all excluded values of 3x²−25.
Show answer
x² − 25 = (x − 5)(x + 5) = 0 ⇒ x = 5 or x = −5.
Find all excluded values of xx(x−4).
Show answer
x(x − 4) = 0 ⇒ x = 0 or x = 4. (Both are forbidden even though the top also has an x — you find excluded values from the bottom, before any cancelling.)
Does 2xx²+9 have any excluded values?
Show answer
No. x² + 9 = 0 would need x² = −9, impossible for a real x. The bottom is never 0, so the no-go list is empty.
For what value of x is x+6x−1 equal to 0?
Show answer
Set the top x + 6 = 0 ⇒ x = −6. Check the bottom: −6 − 1 = −7 ≠ 0, so −6 is allowed. The expression is 0 at x = −6.
Is x+4x²−16 equal to 0 when x = −4?
Show answer
No — it’s undefined there. The top is −4 + 4 = 0, but the bottom is also (−4)² − 16 = 0. Both zero ⇒ 0 ÷ 0 ⇒ undefined, not a zero.
Evaluate x−3x+1 at x = 5, and state its verdict at x = −1.
Show answer
At x = 5: 5−35+1 = 26 = ⅓. At x = −1: bottom −1 + 1 = 0 ⇒ undefined.
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?
Show answer
Cost each = 48n, excluded value n = 0 (you can’t split a bill among nobody). At n = 6: 486 = $8 each.
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.
Show answer
Time = 150v hours, excluded value v = 0 (a stopped car never arrives). At v = 50: 15050 = 3 hours.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
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.
eastmath.com · Stage 9 · 9.1 Meeting the Rational Expression · Intuition before notation
eastmath.com · 9.1 Meeting the Rational Expression · 9.1.1 What is a rational expression?