Factoring overview

Factoring is running multiplication backwards. To factor the integer 253, you consider how you could get it by multiplying smaller integers; the answer is that 253 = 11 × 23. Similarly, to factor the polynomial 2x2 + 5x + 3, you consider how you could get it by multiplying simpler polynomials; the answer is that 2x2 + 5x + 3 = (x + 1) ⋅ (2x + 3). (We'll discuss later how you could come up with that answer.) As the integers 11 and 23 are the factors of 253, so the polynomials x + 1 and 2x + 3 are the factors of 2x2 + 5x + 3.

The simplest technique for factoring is when one of the factors is a constant. When you do this, here are some rules to follow, which apply whenever the original polynomial has only rational coefficients (otherwise it gets more complicated):

Here are some ways to factor 6x − 12 that violate these rules and so are wrong: All of these equations are identities (they're always true), but they are not the proper way to factor 6x − 12. The only proper way, following all of the rules above, is 6x − 12 = 6(x − 2).

Here are some more examples:

Sometimes the textbook is a little sloppy about fractional and negative coefficients, and it's true that the rules are somewhat arbitrary; but you need to pick some rules and follow them consistently in order to guarantee that everything will work out in a complicated problem.
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