The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: SAT-II, AMC-10, AMC-12, AIME, Math Kangaroo 9-10.

Reciprocal Equations are equations with symmetric coefficients. For example, a cubic one looks like this:

equation

and a quartic one looks like this:

equation

Properties

1. A reciprocal equation of degree n cannot have a root of zero.

Proof: if it had a root of zero, then the free term would be zero, which would make the leading coefficient also zero. As a result, the equation would not be degree n anymore. Since n is an arbitrary positive integer, the reasoning can be applied recursively to show that the reciprocal equation with a root of zero is identically null.

2. If a reciprocal equation has the root r, then it also has the root 1/r. Proof:

If:

equation

Then:

equation

equation

3. Any odd-degree reciprocal equation has a root of -1. Proof:

equation

equation

Hopefully, these properties will help downgrade the degree of the equation.

Polynomial Equations with Real Coefficients

If a polynomial equation has real coefficients and admits a complex-imaginary root a+ib with b≠0, then it also admits the root a-ib.

Observation: the complex-conjugate roots a+ib and a-ib have the same multiplicity (i.e. if one is a double root, so is the other).

Observation: Any odd-degree polynomial equation with real coefficients has at least one real root.

Observation: Any polynomial equation with real coefficients has an even number of complex-imaginary roots.

Polynomial Equations with Rational Coefficients

If a polynomial equation has rational coefficients and admits an irrational root a+√b such that:

equation

then it also admits the 'conjugate' root a-√b.

Observation: the 'conjugate' roots a+√b and a-√b have the same multiplicity (i.e. if one is a double root, so is the other).

Observation: Any polynomial equation with rational coefficients has an even number of irrational roots.

Rational Root Theorem

If a polynomial equation has integer coefficients and admits a root:

equation

and p, q are coprime:

equation

then p must be a divisor of the free term and q must be a divisor of the leading coefficient.