Multiplying rational expressions looks like the hardest thing in the chapter and is secretly the easiest — if you factor before you do anything else. Factoring turns every top and bottom into a product, and once everything is a product, matching factors pair off and become 1. Step through five problems and watch each pair leave the board.
The factor board
a factor still in the product divided out — it made a 1 survives into the answer
Check it with a number
test valuex = 1
original—
simplified—
Five productspick one, then press Next
Why cancelling is allowed at all. Crossing out a factor is not magic — it is division. (x + 5) divided by (x + 5) is 1, and multiplying by 1 changes nothing. But division only reaches things that are multiplied. That is the entire reason we factor first: factoring is what turns a sum into a product so that division can get at it.
The one fatal error. Never cancel a term. In (x + 5)/x you may not cross out the x’s — the top is a sum, not a product. Test it at x = 5: the true value is 10/5 = 2, but “cancelling” would claim 5. If a factor is not multiplied against everything above the bar, it cannot leave.