Welcome to our community

Be a part of something great, join today!

Solving a quartic polynomial

  • Thread starter
  • Admin
  • #1

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,707
Solve \(\displaystyle x^4+1=2x(x^2+1)\).
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I would first write the quartic in standard form and assume it has two quadratic factors:

\(\displaystyle x^4-2x^3-2x+1=(x^2+ax+1)(x^2+bx+1)=x^4+(a+b)x^3+(2+ab)x^2+(a+b)x+1\)

Equating coefficients, we find:

\(\displaystyle a+b=-2\)

\(\displaystyle 2+ab=0\)

We have two solutions, but they are symmetric, hence we may take:

\(\displaystyle a=\sqrt{3}-1,\,b=-(\sqrt{3}+1)\)

Hence:

\(\displaystyle x^4-2x^3-2x+1=(x^2+(\sqrt{3}-1)x+1)(x^2-(\sqrt{3}+1)x+1)=0\)

Application of the quadratic formula on the two factors gives us:

\(\displaystyle x=\frac{1-\sqrt{3}\pm i\sqrt{2\sqrt{3}}}{2},\,\frac{1+\sqrt{3}\pm\sqrt{2\sqrt{3}}}{2}\)
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,135
You could try this. But I'll admit that I've never gotten through one of those without making any mistakes. It is, in a word, "tedious."

-Dan
 
Last edited:
  • Thread starter
  • Admin
  • #4

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,707
You could try this. But I'll admit that I've never gotten through one of those without making any mistakes. It is, in a word, "tedious."

-Dan
Hi Dan,

The link doesn't work...:eek:

And my solution is less clever than the approach of MarkFL. (Nerd)

First, I let \(\displaystyle x=\tan \theta\) to transform the original equation and making it as \(\displaystyle \sin^2 2\theta+2\sin 2\theta-2=0\).

I then use the quadratic formula to solve for \(\displaystyle \sin 2\theta\) then \(\displaystyle \tan \theta\) to obtain the numerical approximation of the answers where

\(\displaystyle x=\tan 23.5293^{\circ}=0.4354\) and \(\displaystyle x=\tan 66.4707^{\circ}=2.2966\).

Also, I was not able to find the complex roots as well!:eek::eek::eek:
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,135