What may be cancelled
(x + 3)/3 becomes x by striking out the two 3s. Striking out means crossing out the same thing above and below. Is that allowed? Judge five strikings-out.
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(x² − 9) / (x + 3).(x² − 9) / (x + 3).- Factor the top:
(x − 3)(x + 3) / (x + 3). (x + 3)multiplies the whole top. It cancels.- Left with
x − 3. - Compare:
(x + 3) / 3. The 3 is added to x, not a factor. - It does not cancel.
The rule
No. In (x + 3)/3 the 3 on top is added to x. It is not a factor.
Strike out only a shared factor. A factor multiplies the whole top or the whole bottom. A term that is added on is not a factor.
Test the claim at x = 3. (x + 3)/3 gives 6/3 = 2. But x gives 3. The numbers differ, so the rewrite is wrong.
| Written as | Struck out | Before | After | Fair? |
|---|---|---|---|---|
(x + 3)/3 → x | the two 3s | 2 | 3 | no |
(3x + 6)/3 → x + 2 | a 3 from every term | 5 | 5 | yes |
(x² − 9)/(x + 3) → x − 3 | the bracket (x + 3) | 0 | 0 | yes |
(x + 5)/(x + 7) → 5/7 | the two x terms | 4/5 | 5/7 | no |
In (3x + 6)/3, every term on top has a 3. So the top is 3(x + 2). That 3 multiplies the whole top, so it may be struck out.
When the bracket is zero
A struck-out bracket must not be zero. At x = −3 the bottom of (x² − 9)/(x + 3) is 0. Then the fraction has no value. So x − 3 matches the fraction only where x + 3 is not zero.