Factoring overview

There is an analogy between polynomials and the way we write integers in base 10. For example, the integer 253 is like the polynomial 2x² + 5x + 3 (especially if you evaluate it when x = 10). When you add integers, you can arrange the digits in columns and add the columns; when you add polynomials, you can arrange the terms in columns and add the columns. When you multiply integers, you need to multiply each digit in one integer by each digit in the other; when you multiply polynomials, you need to multiply each term in one polynomial by each term in the other. (In some ways, arithmetic with polynomials is easier to understand, since there is no carrying.)

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):

There can only be one constant factor, which comes first.
In each non-constant factor, the leading coefficient should be positive.
In each non-constant factor, all of the coefficients should be integers.
In each non-constant factor, the coefficients should not have any common integer factor greater than 1.

Here are some ways to factor 6x − 12 that violate these rules and so are wrong:

6x − 12 = 2(x − 2)3;
6x − 12 = −6(−x + 2);
6x − 12 = 12(½x − 1);
6x − 12 = 2(3x − 6).

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:

Look for a factor common to each term: 2y + 20 = 2 ⋅ y + 2 ⋅ 10 = 2(y + 10).
Sometimes it helps to factor the coefficients: 6t + 15 = 2 ⋅ 3 ⋅ t + 3 ⋅ 5 = 3(2t + 5).
You can also try to factor out a variable that appears in every term: 5x³ + 3x² = 5xxx + 3xx = x²(5x + 3).
A negative coefficient on the leading term means that a minus sign appears out front: 8 − 2x = −2x + 8 = −2x − −8 = −2(x − 4).
If the coefficients are fractional, then their common denominator appears in the denominator out front: ½x + ⅔ = 3/6 ⋅ x + 4/6 = ⅙(3x + 4).

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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This web page was written between 2012 and 2016 by Toby Bartels, last edited on 2016 July 13. Toby reserves no legal rights to it.

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