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
2*x*^{2} + 5*x* + 3,
you consider how you could get it by multiplying simpler polynomials;
the answer is that 2*x*^{2} + 5*x* + 3 =
(*x* + 1) ⋅ (2*x* + 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 2*x* + 3
are the *factors* of 2*x*^{2} + 5*x* + 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.

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

Here are some more examples:

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

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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. The permanent URI of this web page is

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