The same fraction rules as Stage 3 — but you must factor first, and the holes stay.
Point 5 of 5 in this lesson: 9.2.5 Building up to the common denominator
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.
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.
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.
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.
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−2 still 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.
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.
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) |
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.
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.
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) |
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), not x(x+1)². Take each distinct factor to its highest power — once.
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.
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.
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.
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.
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?
Reduce to lowest terms: 8a³b12ab².
Reduce: x²−9x+3. State the restriction.
Reduce: 2xx²+x.
Spot the error. A student writes x+55 = x. Is this right? Check with x = 10.
Reduce: x²−4x+4x²−4.
Hidden hole. Reduce x²−4x²−5x+6 and state every restriction.
Find the LCD of …9x² and …6x.
Find the LCD of …x²−1 and …x−1.
Build both fractions onto their LCD: 5x and 3x+1.
Challenge. Build both onto their LCD: 7x²−1 and 2x+1.
Six questions to lock it in. Tap the answer you think is right.
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.