Editing Factorization (section)

====Adding and subtracting terms====
Sometimes, some term grouping reveals part of a [[#Recognizable patterns|recognizable pattern]]. It is then useful to add and subtract terms to complete the pattern.

A typical use of this is the [[completing the square]] method for getting the [[quadratic formula]].

Another example is the factorization of <math>x^4 + 1.</math> If one introduces the non-real [[square root of –1]], commonly denoted {{mvar|i}}, then one has a [[difference of squares]]
<math display="block">x^4+1=(x^2+i)(x^2-i).</math>
However, one may also want a factorization with [[real number]] coefficients. By adding and subtracting <math>2x^2,</math> and grouping three terms together, one may recognize the square of a [[binomial (polynomial)|binomial]]:
<math display="block">x^4+1 = (x^4+2x^2+1) - 2x^2 = (x^2+1)^2 - \left(x\sqrt2\right)^2 = \left(x^2+x\sqrt2+1\right) \left(x^2-x\sqrt2+1\right).</math>
Subtracting and adding <math>2x^2</math> also yields the factorization:
<math display="block">x^4+1 = (x^4-2x^2+1)+2x^2 = (x^2-1)^2 + \left(x\sqrt2\right)^2 = \left(x^2+x\sqrt{-2}-1\right) \left(x^2-x\sqrt{-2}-1\right).</math>
These factorizations work not only over the complex numbers, but also over any [[field (mathematics)|field]], where either –1, 2 or –2 is a square. In a [[finite field]], the product of two non-squares is a square; this implies that the [[polynomial]] <math>x^4 + 1,</math> which is [[irreducible polynomial|irreducible]] over the integers, is reducible [[modular arithmetic|modulo]] every [[prime number]]. For example,
<math display="block">x^4 + 1 \equiv (x+1)^4 \pmod 2;</math>
<math display="block">x^4 + 1 \equiv (x^2+x-1)(x^2-x-1) \pmod 3,</math>since <math>1^2 \equiv -2 \pmod 3;</math>
<math display="block">x^4 + 1 \equiv (x^2+2)(x^2-2) \pmod 5,</math>since <math>2^2 \equiv -1 \pmod 5;</math>
<math display="block">x^4 + 1 \equiv (x^2+3x+1)(x^2-3x+1) \pmod 7,</math>since <math>3^2 \equiv 2 \pmod 7.</math>