Roots of polynomial equations ( Substitution )

[Erfan]

How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?

 

[M R]

How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?

Since \(\displaystyle u \ne 0\) you may divide through by \(\displaystyle u^2\).

What do you notice now?

 

[caffeinemachine]

How do I reduce u^4 + 5u^3 + 6u^2 + 5u + 1 = 0 to v^2 + 5v + 4 = 0 by using v = u + 1/u ?

You can do as M R suggested or you can put $v=u+1/u$ in $v^2+5v+4=0$. You should get $u^4 + 5u^3 + 6u^2 + 5u + 1 = 0$.

 

[M R]

You can use the same idea to solve \(\displaystyle x^6-6x^5+14x^4-18x^3+14x^2-6x+1=0\), which I made specially for you.

 

[M R]

And a slightly nicer one \(\displaystyle x^6-9x^5+29x^4-42x^3+29x^2-9x+1=0\).