Solution - Finding the roots of polynomials

Step  1  :

Polynomial Roots Calculator :

 1.1    Find roots (zeroes) of :       F(w) = w⁵-1
Polynomial Roots Calculator is a set of methods aimed at finding values of  w  for which   F(w)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  w  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  -1.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1

 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   w⁵-1 
can be divided with  w-1 

Polynomial Long Division :

 1.2    Polynomial Long Division
Dividing :  w⁵-1 
                              ("Dividend")
By         :    w-1    ("Divisor")

Quotient :  w⁴+w³+w²+w+1  Remainder:  0 

Polynomial Roots Calculator :

 1.3    Find roots (zeroes) of :       F(w) = w⁴+w³+w²+w+1

     See theory in step 1.1
In this case, the Leading Coefficient is  1  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  (w4 + w3 + w2 + w + 1) • (w - 1)