Factoring as Reverse Multiplication

Algebra 1 · Axiom Academy

Factoring as Reverse Multiplication Multiplying factors builds a polynomial. Factoring runs that same machine backwards — to recover the factors it came from. One machine, run two directions You already know how to multiply two binomials together: (x+2)(x+3) opens up into the polynomial x^2 + 5x + 6 . Factoring is not a brand-new skill — it is that exact process played in reverse. Watch it run forward, then watch the arrow flip and the polynomial fold right back into the factors it came from. First the factors multiply out: each piece of (x+2) meets each piece of (x+3) , the four products land, and like terms combine into the polynomial. Then the direction reverses — the polynomial collapses back into (x+2)(x+3) . Forward is multiplying; backward is factoring. Multiplying and factoring are the same road travelled in opposite directions. Where the polynomial comes from: an area box Pick the two factors with the sliders. The box is (x+a) across the top and (x+b) down the side — its four tiles are the four products, and their total area is the polynomial. Notice the middle term is always a+b and the last term is always a b . Hold onto that — it is the key to going backwards. Multiplying is easy: the box always fills the same way, and the polynomial falls out. Factoring: hunt for the pair that fits

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