Rational expressions, step by step

A rational expression simplifies only when the top and the bottom share a factor.

The question

Simplify (x² + 12x − 160) / (x² − 15x + 56)

(x² + 12x − 160) / (x² − 15x + 56)

①x² + 12x − 160 = (x − 8)(x + 20)

②x² − 15x + 56 = (x − 8)(x − 7)

①/②

Nothing can cancel while these are sums. Factor the top first — two numbers that multiply to −160 and add to 12. Call it ①.
Now the bottom, the same way — call it ②. (x − 8) turns up again, and that is not a coincidence: it is the whole reason this expression simplifies.
The question is ① over ②. (x − 8) is a FACTOR of both, so it divides out and leaves 1. Only a factor can go this way — striking the x² off the top and the bottom of the sums above is the mistake this page exists to catch. And the cancelling changed something quietly: x may not be 7 or 8, because the original has no value there.

How it works.

01
Nothing can cancel while these are sums. Factor the top first — two numbers that multiply to −160 and add to 12. Call it ①.
02
Now the bottom, the same way — call it ②. (x − 8) turns up again, and that is not a coincidence: it is the whole reason this expression simplifies.
03
The question is ① over ②. (x − 8) is a FACTOR of both, so it divides out and leaves 1. Only a factor can go this way — striking the x² off the top and the bottom of the sums above is the mistake this page exists to catch. And the cancelling changed something quietly: x may not be 7 or 8, because the original has no value there.