Factor both parts, then cancel whole factors — never single terms.
Simplify
x² − 9x² + 7x + 12
Factor both: (x + 3)(x − 3) over (x + 3)(x + 4). The (x + 3) cancels, leaving x − 3x + 4. A cancels the x² terms, which is the one thing you may never do — x² is a term, not a factor. C cancels the wrong bracket. Test with x = 1: the original is −820 = −0.4, and 1 − 31 + 4 = −0.4 ✓
The denominator can never be zero, so some values of x are excluded.
x + 5x² − 4
find the restrictions
x² − 4 = (x + 2)(x − 2)
factor the bottom
x ≠ 2 and x ≠ −2
both roots excluded
If a factor cancels later, its restriction still applies. In the opening problem x ≠ −3 even though (x + 3) disappeared — the original expression was never defined there.
A factor is something the whole expression is multiplied by. In x + 2 the x is being added to 2, so it is not a factor of anything.
Test any doubtful cancellation with a number. If x + 2x + 5 could become 25, then at x = 1 it would be 36 = 0.5 and 0.4 at once. It cannot.
2x² + 6xx² + x − 6
nothing cancels yet
top: 2x(x + 3)
GCF first
bottom: (x + 3)(x − 2)
product −6, sum 1
= 2xx − 2
x ≠ −3, x ≠ 2
(a − b) and (b − a) differ only by a sign.
x − 55 − x
nearly the same, reversed
5 − x = −(x − 5)
factor out −1
= x − 5−(x − 5) = −1
now it cancels
Check with x = 7: 7 − 55 − 7 = 2−2 = −1 ✓ This pattern turns up constantly once subtraction gets involved.
"Already in lowest terms" is a complete and correct answer. Inventing a cancellation to make the expression look tidier is how the term-cancelling error starts.
Exponents subtract, and what is left sits on the side that had more.
Simplify 15a²20a⁶.
1520 = 34
numbers: divide by 5
a²a⁶ = 1a⁴
four more a's on the bottom
= 34a⁴
a ≠ 0
The a's do not vanish just because the top ran out — the extra a⁴ stays below the line.
Simplify 5x − 1015.
= 5(x − 2)15
factor the top first
= x − 23
cancel the 5 with a third of 15
Cancelling 5x with 15 and leaving the 10 gives x − 103. Test with x = 5: the original is 1515 = 1, but 5 − 103 = −53. The 5 must come out of both terms before it can cancel.
Factor, then cancel — even when the factors look disguised.
6 − xx² − 36
= −x − 6(x + 6)(x − 6)
6 − x = −(x − 6)
= −1x + 6
x ≠ 6, x ≠ −6
An opposite bracket leaves a −1 behind, which goes in front of the answer.
x² − y²x² + xy
= (x + y)(x − y)x(x + y)
squares on top, GCF x below
= x − yx
x ≠ 0, x ≠ −y
With two letters the restriction can involve both: the bottom is zero whenever x = 0 or x = −y.
2x² + 5x − 34x² − 1
top: (2x − 1)(x + 3)
ac = −6: 6 and −1
bottom: (2x + 1)(2x − 1)
difference of squares
= x + 32x + 1
x ≠ 12, x ≠ −12
The excluded values can be fractions: 2x − 1 = 0 when x = 12.
Items ask for the simplest form, for the values x cannot take, or for which cancellation is allowed. The fastest route is to factor top and bottom completely, cancel whole brackets, and read the restrictions off the original bottom. For a 10-second check, put an allowed number such as x = 2 into the question and your answer — they must agree.
Simplify
x² − 4xx² − 16
Top: x(x − 4). Bottom: (x + 4)(x − 4). The (x − 4) cancels, leaving xx + 4, with x ≠ 4 and x ≠ −4. A cancels the x² terms, which is never allowed. D forgets the x that was factored out of the top. Test x = 2: original is −4/−12 = 13, and 22 + 4 = 13 ✓
Hint: the top needs a GCF taken out, the bottom is a difference of squares. Factor both before looking for anything to cancel.
