Editing Monic polynomial (section)

==Polynomial equations==

Let <math>P(x)</math> be a [[polynomial equation]], where {{mvar|P}} is a [[univariate polynomial]] of degree {{mvar|n}}. If one divides all coefficients of {{mvar|P}} by its [[leading coefficient]] <math>c_n,</math> one obtains a new polynomial equation that has the same solutions and consists to equate to zero a monic polynomial.

For example, the equation
:<math>2x^2+3x+1 = 0</math>
is equivalent to the monic equation
:<math>x^2+\frac{3}{2}x+\frac{1}{2}=0.</math>

When the coefficients are unspecified, or belong to a [[field (mathematics)|field]] where division does not result into fractions (such as <math>\R, \Complex,</math> or a [[finite field]]), this reduction to monic equations may provide simplification. On the other hand, as shown by the previous example, when the coefficients are explicit integers, the associated monic polynomial is generally more complicated. Therefore, [[primitive polynomial (ring theory)|primitive polynomial]]s are often used instead of monic polynomials when dealing with integer coefficients.