Factoring Review: Introduction

Calculus Readiness · Axiom Academy

Multiplying builds a product out of pieces. Factoring runs the film backward — and once you can see the pieces, the answers fall out. Factoring is multiplying, in reverse You already know how to multiply (x+2)(x+3) out into x^2 + 5x + 6 . Factoring is the same picture run backward: you start with the product x^2 + 5x + 6 and recover the two factors that built it. Do that, and solving equations, simplifying fractions, and finding where a function hits zero all become easy — because the answers are sitting right there in the factors. Here is the trick made visible. An expression like x^2 + 5x + 6 is the area of a rectangle. Watch one rectangle of that area get carved by a vertical cut and a horizontal cut into four tidy pieces — and the lengths of its two sides turn out to be exactly the factors (x+3) and (x+2) . The four pieces add to x^2 + 5x + 6 ; the sides multiply to (x+3)(x+2) . Same rectangle — that is what "factored" means. Warm up on a number: what are 12's factors? Before letters, try it with a number you know. Here are 12 dots. Drag the dial to arrange them into rows — whenever they snap into a perfect rectangle with no leftovers, you have found a factor pair of 12 . The numbers that refuse to make any rectangle except a single line are the primes. , and also and — every rectangle is a way to factor it. A prime like 7 only ever makes the flat line. Back to x^2 + 5x + 6 : split the middle term

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