Help Solving a Quartic Polynomial

  • Context: Undergrad 
  • Thread starter Thread starter DuncanM
  • Start date Start date
  • Tags Tags
    Polynomial
Click For Summary
SUMMARY

The discussion centers on solving the quartic polynomial equation derived from the relationship arccos(Y) = arctan(Y), where Y = 1/x, resulting in the equation x4 - x2 - 1 = 0. A participant suggests simplifying the quartic by substituting u = x2, transforming it into a quadratic equation u2 - u - 1 = 0, which can be solved for u. This method effectively allows for finding the roots of the original quartic equation by subsequently solving for x.

PREREQUISITES
  • Understanding of polynomial equations, specifically quartic and quadratic forms.
  • Familiarity with trigonometric functions and their inverse functions.
  • Knowledge of substitution methods in algebra.
  • Ability to solve quadratic equations using the quadratic formula.
NEXT STEPS
  • Study the quadratic formula and its applications in solving equations.
  • Explore polynomial long division for simplifying higher-degree polynomials.
  • Learn about the properties of inverse trigonometric functions and their relationships.
  • Investigate numerical methods for solving polynomials when analytical solutions are complex.
USEFUL FOR

Mathematicians, students studying algebra, and anyone interested in solving polynomial equations analytically.

DuncanM
Messages
99
Reaction score
3
I am trying to solve an equation:

arccos(Y) = arctan(Y), where Y = 1/x

This turns into a quartic equation:

x4 - x2 - 1 = 0

It looks simple enough to simplify further; however, I must be having a brain-fart because I can't do it. I'd like to avoid resorting to a numerical solution.

Am I overlooking something?
Can this equation be simplified further? If so, how?

Any help is much appreciated.
 
Mathematics news on Phys.org
Setting [itex]u = x^2[/itex], you can rewrite it as [itex]u^2 - u - 1 = 0[/itex]. You can solve this easily enough for the roots in u, and then solve the roots for x.
 
Excellent!

Thank-you very much!
 

Similar threads

Undergrad Factoring a quartic polynomial over the reals
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Proof for interpolating polynomial and error approximation
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Eccentric anomaly of ellipse-circle intersections
  • · Replies 4 ·
Replies
4
Views
2K
MHB Direct solution for two unknowns in two equations
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
MHB Understanding Cubic Equation Formula for Polynomials of Degree Three
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad Polynomial Division: Solve with Mike's Help!
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Finding Critical Points of a Quartic Function: A Scientific Approach
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Quartic eqn solutions for Cos/Sin
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Can De Moivre's Theorem Simplify Solving Complex Polynomial Equations?
  • · Replies 3 ·
Replies
3
Views
2K