How can I solve this: x^4+x^3+x^2+x+1=0   ? thx

1 Answer | Add Yours

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

To solve x^4+x^3+x^2+x+1 = 0.

We know that 1+x+x^2+x^4 =  (x^5-1)/(x-1).

So the given equation is therefore rewritten as:

(x^5-1) /(x-1) = 0

Multiply by (x-1). So , mind x-1 is our factor.

(x^5 -1) = 0

x^5 = 1.

We know 1 = cos2npi +isin2npi.

Therefore ,

x^5 =  cos2npi+isin2npi

Take the 5 th root.

x = (co2npi+isin2npi) ^(1/5)

x = (cos(2npi)/5 +isin(2npi)/5 , for n = 0,1,2,3,4....by D'Moivres theorem.Actually after  n=4, for the next integral values the roots repeat.

x0 = 1  is not the root as we have multiplied by our factor (x-1) to the given expression (x^4+x^2+x^3+1 ).

x1 = cos72+isin72

x2 = cos144 + isin 144

x3 = cos216 +i sin 216

x4  = cos288 +isin288.

So x1,x2,x3 and x4 are the solutions.

 

We’ve answered 318,911 questions. We can answer yours, too.

Ask a question