Editing Cubic equation (section)

==Factorization==
If the coefficients of a cubic equation are [[rational number]]s, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a [[common multiple]] of their denominators. Such an equation 
<math display="block">ax^3+bx^2+cx+d=0,</math>
with integer coefficients, is said to be [[reducible polynomial|reducible]] if the polynomial on the left-hand side is the product of polynomials of lower degrees. By [[Gauss's lemma (polynomial)|Gauss's lemma]], if the equation is reducible, one can suppose that the [[Factorization|factors]] have integer coefficients.

Finding the roots of a reducible cubic equation is easier than solving the general case. In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form
<math display="block">qx-p,</math>
with {{mvar|q}} and {{mvar|p}} being [[coprime integers]]. The [[rational root test]] allows finding {{mvar|q}} and {{mvar|p}} by examining a finite number of cases (because {{mvar|q}} must be a divisor of {{mvar|a}}, and {{mvar|p}} must be a divisor of {{mvar|d}}).

Thus, one root is <math>\textstyle x_1= \frac pq,</math> and the other roots are the roots of the other factor, which can be found by [[polynomial long division]]. This other factor is 
<math display="block">\frac aq\,x^2+ \frac{bq+ap}{q^2}\,x+\frac{cq^2+bpq+ap^2}{q^3}.</math>
(The coefficients seem not to be integers, but must be integers if {{tmath|p/q}} is a root.)

Then, the other roots are the roots of this [[quadratic polynomial]] and can be found by using the [[quadratic formula]].