Polynomial Equations and Symmetric Functions
While algorithms for solving polynomial equations of
degree at most 4 exist, there are in
general no such algorithms for polynomials of higher degree. A polynomial equation to be
solved at an Olympiad is usually solvable by using the Rational Root Theorem (see the
earlier handout Rational and irrational numbers), symmetry, special forms, and/or
Here are, for the record, algorithms for solving 3rd and 4th degree equations.
Algorithm for solving cubic equations. The general cubic equation
can be transformed (by dividing by and letting ) into an equation of the form
To solve this equation, we substitute x = u + v to obtain
Note that we are free to restrict u and v so that . Then and are the roots
of the equation . Solving this equation, we obtain
Now we may choose cube roots so that
Then A + B is a solution. It is easily checked that the other pairs are obtained by rotating
A and B in the complex plane by angles . So, the full set of solutions is
Ferrari's method of solving quartic equations. The general quartic equation is reduced
to a cubic equation called the resolvent. write the quartic equation as
Transpose to obtain
and then adding to
both sides makes the left-hand side equal to .
If r can be chosen to make the right-hand side a perfect square, then it will be easy to find
all solutions. The right-hand side,
is a perfect square if and only if its discriminant is zero. Thus we require
This is the cubic resolvent.
Reciprocal or palindromic equations. If the equation the form
and for all j = 0,... ,n, it is called palindromic. For even n, the transformation
reduces the equation to a new one of degree n/2. After finding all solutions ,
the solutions of the original equation are found by solving quadratic equations .
1. Solve .
2. Solve .
3. Solve .
4. Solve .
5. Solve and find values of a for which all roots are real.
Definitions. A function of n variables is symmetric if it is invariant under any permutation
of its variables. The kth elementary symmeric function is defined by
where the sum is taken over all choices of the indices from the set
Symmetric function theorem. Every symmetric polynomial function of is a
polynomial function of . The same conclusion holds with "polynomial" replaced
by "rational function".
Theorem. Let be the roots of the polynomial equation
and let be the kth elementary symmetric function of . Then
Newton's formula for power sums. Let
where are the roots of
1. Find all solutions of the system
determine the value of .
3. Let , where a, b, c, A, B, C are real numbers
and A+ B+ C is a multiple of π. Prove that if , then for all k ∈ IN.
4. Find a cubic equation whose roots are the cubes of the roots of .
5. Find all values of the parameter a such that all roots of the equation
6. A student awoke at the end of an algebra class just in time to hear the teacher say,
"...and I give you a hint that the roots form an arithmetic progression." Looking at the
board, the student discovered a fifth degree equation to be solved for homework, but
he had time to copy only
before the teacher erased the blackboard. He was able to find all roots anyway. What
are the roots?