Reducing a rational expression in one picture: factor, cancel matching factors, and remember the hole the cancelled factor left behind.
Stage 3 taught you that 2/4 and 1/2 are the
same number wearing different clothes. You shrank fractions by dividing top and bottom by a common
factor, and you matched bottoms before adding. Every one of those moves still works here. The
only twist: now the top and bottom are polynomials, so before you can spot a common factor
you must factor first — and because the bottom hides a letter, the
excluded values from 9.1
do not disappear when a factor cancels. A restriction the original bottom forced can outlive — and even be
hidden by — the very factor that cancels, so you must carry it along by hand.
By the end of this lesson you'll be able to: use the fundamental property to rewrite a
fraction without changing its value; reduce a rational expression to lowest terms by
factoring and cancelling whole factors; spot and refuse the lesson's famous trap
(cancelling terms); and build several fractions onto one shared bottom — the
least common denominator — which is the launch pad for adding and subtracting in
9.4. Throughout, we keep one
color code: the numerator (top) is amber, the
denominator (bottom) is blue, a
cancelled common factor is green, and an
excluded value is red.
9.2.1 The fundamental property
Here is the engine that drives the whole lesson. If you multiply the top and the bottom of a
fraction by the same nonzero amount, the fraction's value does not change. You already trust
this for numbers: 1/2 = 2/4 = 3/6. Each step multiplies top and bottom by the
same thing — and you can run it backwards to shrink a fraction, too.
Key idea — the fundamental property
AB
=
A·MB·M( M ≠ 0 )
Scaling top and bottom by the same nonzero M leaves the value untouched. Read left-to-right
it builds up; read right-to-left it reduces. The catch unique to Stage 9: if M
is a letter expression, it must not be 0 either.
The same property with a letter in the basement:
2x
becomes
2(x+1)x(x+1)
when we choose M = x+1. The value at any allowed x is
identical — we only dressed it up so its bottom matches some other fraction's. That "dressing up" move
is exactly how we'll force a common denominator in §9.2.4.
Numbers on the left, letters on the right — the move is identical. Multiply top and bottom by the same nonzero thing.
🎮 Try itFUNDAMENTAL-PROPERTY SCALER
Pick a fraction and a multiplier M. Watch top and bottom both get ·M — and watch the value stay put.
Fraction
Multiply top & bottom by M =
9.2.2 Reducing — factor first, then cancel factors
To shrink a rational expression you run the fundamental property backwards: find a factor common to
the whole top and the whole bottom, and divide it
out. The one new rule, and it is non-negotiable: factor first. A polynomial like
x²−4 doesn't look like it shares anything with
x²−4x+4 until you write both as products.
Worked example — numbers and letters
Monomial fraction.6x²y9xy²
: the numbers give 6/9 = 2/3; the x²/x leaves one
x on top; the y/y² leaves one y
on the bottom. Result:
2x3y.
Factor, then cancel.x²−4x²−4x+4
=
(x−2)(x+2)(x−2)(x−2).
The factor (x−2) appears on top and bottom, so it cancels once, leaving
x+2x−2.
That last result is beautiful but it hides a debt. The original expression was undefined at
x = 2 (the bottom x²−4x+4 = (x−2)² is 0 there) and also
at... well, only at 2. After cancelling, the new form
x+2x−2still blows up at x = 2, so nothing is lost there. But cancelling can
also erase a hole from view. The reduced form is only equal to the original on the original's
domain: the restriction x ≠ 2 rides along even after the
(x−2) is gone.
Watch — the #1 trap of this whole lesson
You may cancel a factor (something multiplied across the whole top and the
whole bottom). You may never cancel a term — a piece sitting beside a + or −.
x+3x
≠ 3 (can't cancel the x)
·
x+3x+5
≠
35(the x's are not factors)
Test it with a number: at x = 4,
x+3x
= 7/4 = 1.75, not 3. The "cancel the x" move is simply false.
🎮 Try itCANCEL-ONLY-FACTORS MACHINE
Top and bottom are already factored into blocks. Click a matching pair of factors to cancel them (they turn green). Then try the trap.
9.2.3 Lowest terms — knowing when to stop
A rational expression is in lowest terms when the numerator and
denominator share no remaining common factor — there is simply nothing
left to cancel. That is the finish line of every "simplify" instruction. The reliable test is the same
every time: factor both completely; if no factor appears in both lists, you're done.
Expression
Factored
Lowest terms?
x²−9x+3
(x−3)(x+3) / (x+3)
No → reduce to x−3
x+2x−2
(x+2) / (x−2)
Yes ✓
2xx²+x
2x / x(x+1)
No → reduce to 2/(x+1)
Key idea
Lowest terms is a property of the factored forms, not of how big the expression looks.
(x+2)/(x−2) can't shrink even though both halves contain an x — because the
x's are tangled up in sums, not standing alone as factors.
One subtle reward for reaching lowest terms: it makes adding and subtracting
far lighter later, and it keeps numbers small when you finally substitute a value for
x.
🎮 Try itLOWEST-TERMS CHECKER
Step a slider through the cancelling, factor by factor. Stop when the green "lowest terms" badge lights — and never before.
Expression
Cancel step
0
9.2.4 The least common denominator (LCD)
To add, subtract, or compare fractions you need them over one shared bottom — and the
smallest one that works is the least common denominator. The recipe is the same recipe you used
with numbers, just stated in factors: factor every denominator, then take each distinct factor to
its highest power.
Worked example — warm up with numbers, then add letters
Numbers. For 1/6 and 1/4:
6 = 2·3 and 4 = 2². Take the highest power of each prime:
2²·3 = 12. (Not 24 — that works, but it isn't least.)
Letters. For
…6x
and
…4x²:
6x = 2·3·x and 4x² = 2²·x². Highest power of each:
2²·3·x² = 12x².
Denominators
Factored
LCD
6x and 4x²
2·3·x | 2²·x²
12x²
x and x+1
x | (x+1)
x(x+1)
x(x+1) and x+1
x·(x+1) | (x+1)
x(x+1)
x²−1 and x+1
(x−1)(x+1) | (x+1)
(x−1)(x+1)
Watch
Don't just multiply the two denominators together — that often overshoots. Because
x(x+1) already contains the factor (x+1),
the LCD of x(x+1) and x+1 is just
x(x+1), notx(x+1)². Take each distinct factor
to its highest power — once.
🎮 Try itLCD BUILDER
Two denominators, shown as bags of factors. Watch the LCD assemble by taking the union, each factor to its highest power.
Denominator pair
9.2.5 Building up to the common denominator
Finding the LCD is half the job; the other half is rewriting each fraction over it without
changing its value — which is the fundamental property again, run forwards. Ask of each fraction: what
factor is my bottom missing to become the LCD? Multiply top and bottom by exactly that.
Worked example — build both onto x(x+1)
Combine the setup for
5x
and
3x+1.
The LCD is x(x+1).
The first bottom x is missing (x+1):
5x
=
5(x+1)x(x+1).
The second bottom x+1 is missing x:
3x+1
=
3xx(x+1).
Now both wear the same bottom and are ready to combine in 9.4.
Notice we only ever multiplied by missing factors — never by extra ones. If you build a
fraction onto a bottom bigger than the LCD, your arithmetic still works, but you've made the numbers
needlessly large and you'll have to reduce again at the end. Aim straight for the LCD.
🎮 Try itBUILD-UP CHOOSER
Each fraction needs a missing factor to reach the LCD. Pick the right one for each — the widget rewrites it and checks you.
Pair
★ The big ideas, in one breath
A rational expression follows every fraction rule you already know — but you must
factor first. The fundamental property (A/B = A·M/B·M,
M ≠ 0) lets you scale a fraction up to build a common bottom or run it
backwards to reduce. You may cancel only whole factors, never a
term beside a + or − — and a value the original bottom forbade
stays forbidden even after its factor cancels. To put several fractions over one bottom, build
the least common denominator by taking every distinct factor to its highest power, then
build each fraction up by its missing factors.
What's next
With reducing and common denominators in hand you're ready for the operations:
9.3 multiplies and divides
(where you cancel across fractions before multiplying), and
9.4 finally adds and subtracts —
standing entirely on the LCD work you just did. Everything funnels toward
9.5, where ignoring those holes
produces fake "extraneous" roots.
✎ Exercises 9.2
Fill the blank using the fundamental property:
3x
=
?x(x−4).
What did you multiply top and bottom by, and what is the new top?
Show answer
Multiply top and bottom by the missing factor (x−4). New top = 3(x−4), i.e. 3x−12. (Valid for x ≠ 0 and x ≠ 4.)
Reduce to lowest terms:
8a³b12ab².
Show answer
8/12 = 2/3; a³/a = a²; b/b² = 1/b. Result: 2a²3b.
Reduce:
x²−9x+3.
State the restriction.
Show answer
x²−9 = (x−3)(x+3). Cancel (x+3): result x−3, with the restriction x ≠ −3 carried along (the original bottom was 0 there).
Reduce:
2xx²+x.
Show answer
x²+x = x(x+1). Cancel the factor x: 2x+1, with x ≠ 0 and x ≠ −1.
Spot the error. A student writes
x+55= x. Is this right? Check with x = 10.
Show answer
Wrong — that cancels a term, not a factor. At x = 10: 15/5 = 3, not 10. The 5 in x+5 is being added, so it can't cancel. The expression is already in lowest terms.
Reduce:
x²−4x+4x²−4.
Show answer
x²−4x+4 = (x−2)² and x²−4 = (x−2)(x+2). Cancel one (x−2): x−2x+2, with x ≠ 2 and x ≠ −2. (This is the hero figure, flipped.)
Hidden hole. Reduce
x²−4x²−5x+6
and state every restriction.
Show answer
x²−4 = (x−2)(x+2) and x²−5x+6 = (x−2)(x−3). Cancel (x−2): x+2x−3. The reduced bottom only shows x ≠ 3 — but the cancelled (x−2) hid a second restriction. You must still write x ≠ 2 as well, or the two forms aren't equal at x = 2.
Find the LCD of
…9x²
and
…6x.
Show answer
9x² = 3²·x², 6x = 2·3·x. Highest power of each: 2·3²·x² = 18x².
Find the LCD of
…x²−1
and
…x−1.
Show answer
x²−1 = (x−1)(x+1); the second already lives inside it. LCD = (x−1)(x+1) (= x²−1).
Build both fractions onto their LCD:
5x
and
3x+1.
Show answer
LCD = x(x+1). 5x = 5(x+1)x(x+1) and 3x+1 = 3xx(x+1).
Challenge. Build both onto their LCD:
7x²−1
and
2x+1.
Show answer
x²−1 = (x−1)(x+1), so LCD = (x−1)(x+1). The first already has it: 7(x−1)(x+1). The second is missing (x−1): 2x+1 = 2(x−1)(x−1)(x+1).
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson develops CCSS A‑APR.D.6 (rewrite simple rational expressions in different forms;
reduce by factoring) and leans hard on A‑SSE.A.2 (use the structure of an expression — here,
seeing a polynomial as a product of factors so a common factor becomes visible). The
fundamental-property warm-up with numeric fractions extends 7.EE.A.1 (equivalent forms of
expressions). The #1 misconception is cancelling terms instead of factors:
students "simplify" (x+3)/x to 3, or cancel the x's in
(x+3)/(x+5). The antidote, repeated until it's reflex: "you can only cancel a
factor of the whole top and the whole bottom — factor first, then cancel products, never pieces of a
sum." A 30-second numeric check (plug in a value and watch the two sides disagree) demolishes the
bad move on the spot. A second, quieter point worth saying aloud: a cancelled factor leaves a
hole in the domain that survives the cancellation — the reduced form equals
the original only on the original's domain.
eastmath.com · Stage 9 · 9.2 Reducing & Common Denominators · Intuition before notation
eastmath.com · 9.2 Reducing and Common Denominators · 9.2.2 Reducing — factor first, then cancel factors