A rational expression simplifies only when the top and the bottom share a factor, and nothing can cancel while they are still sums. These free printable worksheets are the page where factoring is finally for something.
The wrong move this skill collects is cancelling TERMS instead of FACTORS — striking the x² off the top and the bottom of the unfactored form. Every step on these keys happens after both halves are products, which is the answer to it: there is nothing to cancel until there are brackets.
Why this sheet works
Both quadratics are built from three distinct whole roots, so the cancellation always exists and both halves always factor. A randomly generated pair would fail to factor on almost every draw, which is why a sheet like this cannot be made by picking numbers.
How this one works
One question from this sheet, worked through a step at a time.
(x² + 5x + 4) / (x² + 11x + 10)
①x² + 5x + 4 = (x + 1)(x + 4)
②x² + 11x + 10 = (x + 1)(x + 10)
①/②(x + 1)(x + 4)(x + 1)(x + 10)
=(x + 4)(x + 10)
Nothing can cancel while these are sums. Factor the top first — two numbers that multiply to 4 and add to 5. Call it ①.
Now the bottom, the same way — call it ②. (x + 1) turns up again, and that is not a coincidence: it is the whole reason this expression simplifies.
The question is ① over ②. (x + 1) 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 −10 or −1, because the original has no value there.
