Palindromic polynomial

A balanced equation is a polynomial ( quite rational ) equation whose coefficient sequence is symmetrical. If the coefficients in magnitude symmetrical, but differ on the sign, it is called an antisymmetric equation.

• 3.3.1 Examples

• 3.4.1 example

Definition

A polynomial equation of degree

Is called symmetric if for all. Applies the other hand, the equation is called antisymmetric. The polynomial

Is then also palindromic (English: palindromic polynomial ) or called antipalindromisch.

Properties

It is a polynomial of degree with real or complex coefficients.

Specific strategies and examples

1st degree

The balanced equation first degree has a solution only -1, the antisymmetric equation corresponding to 1.

2nd degree

Balanced 2nd degree equations can be solved using the quadratic formula.

Example

The equation has the mutually reciprocal solutions 1/3 and 3

3rd degree

Since 1 or -1 solution, performs a polynomial division by or into a square ( symmetric ) equation.

Examples

• The left side of equation 3 degree symmetrical with the solutions -1 / 3, -1, and -3 to be by division by, with consequent further solutions.
• The left side of the antisymmetric 3rd degree equation with the solutions -1 / 3, 1 and -3 becomes by dividing by also, with consequent further solutions.

Grade 4

Dividing the equation by and subsequent substitution leads to an equation of second degree in that can be solved with the quadratic formula. The Resubstition in turn leads to two symmetrical equations 2nd degree.

Example

The equation is determined by the said quadratic equation for substitution with the solutions is from 10 /3 and 5/2 for, resulting in the equations and solutions -2; -1 / 2; 1/3 and 3, of the original equation result.

5th degree

Since 1 or -1 solution, performs a polynomial by or to a symmetric equation of degree 4.