1
Nothing cancels until everything is factored.
A sum has no factors to cancel. Only products do.
not allowed
x² − 9x² + 5x + 6
x² is a term, not a factor
allowed
(x + 3)(x − 3)(x + 3)(x + 2)
a whole bracket, top and bottom
The test
Can you draw a bracket round it and see it multiplied by everything else?
Then it is a factor and it can go. If a plus or minus separates it, it stays.
A single term top and bottom is all factors already: cancel the numbers, then
subtract the powers. 8a³b12ab² = 2a²3b — 8 and 12 share 4, a³ ÷ a = a²,
and b ÷ b² leaves b underneath.
2
Factor the top — difference of squares.
x² − 9
x · x and 3 · 3
(x + 3)(x − 3)
Two perfect squares with a minus between them always split into a plus pair and a minus pair.
Other patterns open a top or bottom the same way. Two cubes:
x³ + 27x² − 9 = (x + 3)(x² − 3x + 9)(x + 3)(x − 3) = x² − 3x + 9x − 3.
Four terms, by grouping: xa + xb + 3a + 3ba² − b² = (x + 3)(a + b)(a − b)(a + b)
= x + 3a − b.
3
Factor the bottom — find the pair.
Two numbers that multiply to 6 and add to 5.
| Pair |
Multiply |
Add |
| 1 and 6 | 6 | 7 |
|---|
| 2 and 3 | 6 | 5 |
|---|
x² + 5x + 6 = (x + 3)(x + 2)
4
Now cancel the matching bracket.
(x + 3)(x − 3)
(x + 3)(x + 2)
(x + 3) ÷ (x + 3) = 1
x − 3x + 2
It will not simplify further. x − 3 and x + 2 share no factor, and you cannot
cancel the loose x's.
5
State what x may not be.
Both cancelled brackets still count — they were there in the original.
x + 3 = 0 → x = −3
the bracket you cancelled
x + 2 = 0 → x = −2
still on the bottom
x ≠ −3 and x ≠ −2
The simplified answer looks perfectly happy at x = −3, but the original was
0 ÷ 0 there. Cancelling hid a hole that never really went away.
6
Brackets that are backwards: take out −1.
(2 − x) and (x − 2) look different but are the same bracket with the sign flipped: 2 − x = −(x − 2). Simplify 2 − xx² + x − 6.
x² + x − 6 = (x + 3)(x − 2)
factor the bottom
2 − x = −(x − 2)
turn the top round
−x − 2(x + 3)(x − 2)
now the brackets match
= −1x + 3
the minus survives the cancelling
a − bb − a = −1, not 1
Cancelling 2 − x against x − 2 as if they were equal gives 1x + 3 — the
answer with the wrong sign, and it will be among the choices. The same goes for
9 − x² = −(x² − 9) = −(x − 3)(x + 3). Test with x = 0: 2−6 = −13 ✓.
7
Variations: an answer that is just a number, a leading 2 on both floors, nothing to cancel.
Simplify 3x − 64 − 2x.
3(x − 2) over −2(x − 2)
take out 3 on top, −2 below
= −32, x ≠ 2
x = 0: −6 ÷ 4 = −32 ✓
Different: every x cancels and a constant is left — still with one banned value.
Simplify 4x² − 92x² + x − 6.
4x² − 9 = (2x − 3)(2x + 3)
(2x)² − 3²
2x² + x − 6 = (2x − 3)(x + 2)
ac = −12: pair 4 and −3
2x + 3x + 2, x ≠ 32, x ≠ −2
x = 0: −9 ÷ −6 = 32 ✓
Different: with a leading 2, a banned value can be a fraction — 2x − 3 = 0 at x = 32.
Simplify x² + 9x² + 3x.
x² + 9: a sum of squares, no factors
the top is one block
x² + 3x = x(x + 3)
nothing matches the top
already in simplest form, x ≠ 0, x ≠ −3
—
Different: sometimes the right answer is “it does not simplify”; the bans still have to be stated.
Cancelling the x²
Striking out x² top and bottom gives 93x = 3x. At x = 1 the original is 104 and 3x
is 3 — the x² was a term, not a factor.
8
Backwards: what makes it reduce?
For what values of b can x² + bx + 8x² − 16 be simplified? The bottom is (x − 4)(x + 4), so the top must share one of those brackets.
share (x − 4): top is 0 at x = 4
a factor makes it zero
16 + 4b + 8 = 0 → b = −6
x² − 6x + 8 = (x − 4)(x − 2)
share (x + 4): top is 0 at x = −4
the other bracket
16 − 4b + 8 = 0 → b = 6
x² + 6x + 8 = (x + 4)(x + 2)
b = −6 gives x − 2x + 4 · b = 6 gives x + 2x − 4
If the question names the simplified form, multiply it back up instead: a result of
x − 2x + 4 means the top was (x − 2)(x − 4) = x² − 6x + 8, so b = −6.
On the exam
Simplifying items reward factoring everything first and watching for reversed
brackets like 4 − x. Others ask where the expression is undefined (use the
original bottom, including cancelled brackets) or for its value at a large x —
simplify first, then substitute. A quick x = 0 or x = 1 test catches a lost minus.
✓
Check: try x = 1 in both.
original top
1 − 9 = −8
x² − 9
original bottom
1 + 5 + 6 = 12
x² + 5x + 6
original
−8 ÷ 12 = −⅔
before simplifying
our answer
(1 − 3) ÷ (1 + 2) = −⅔
a match
Remember the flow
Factor · Cancel brackets · State the bans
Terms never cancel — only factors.
