Solution - Finding the roots of polynomials

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2".  2 more similar replacement(s).

Step  1  :

             1
 Simplify   ——
            x2

Equation at the end of step  1  :

                    1
  ((((x3)-(x2))+x)-——)-1
                   x2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  x²  as the denominator :

                    x3 - x2 + x     (x3 - x2 + x) • x2
     x3 - x2 + x =  ———————————  =  ——————————————————
                         1                  x2        

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x³ - x² + x  =   x • (x² - x + 1) 

Trying to factor by splitting the middle term

 3.2     Factoring  x² - x + 1 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 1 = 1 

Step-2 : Find two factors of  1  whose sum equals the coefficient of the middle term, which is   -1 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 x • (x2-x+1) • x2 - (1)     x5 - x4 + x3 - 1
 ———————————————————————  =  ————————————————
           x2                       x2       

Equation at the end of step  3  :

  (x5 - x4 + x3 - 1)    
  —————————————————— -  1
          x2            

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

         1     1 • x2
    1 =  —  =  ——————
         1       x2  

Checking for a perfect cube :

 4.2    x⁵ - x⁴ + x³ - 1  is not a perfect cube

Trying to factor by pulling out :

 4.3      Factoring:  x⁵ - x⁴ + x³ - 1 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  x³ - 1 
Group 2:  x⁵ - x⁴ 

Pull out from each group separately :

Group 1:   (x³ - 1) • (1)
Group 2:   (x - 1) • (x⁴)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 4.4    Find roots (zeroes) of :       F(x) = x⁵ - x⁴ + x³ - 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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
   x⁵ - x⁴ + x³ - 1 
can be divided with  x - 1 

Polynomial Long Division :

 4.5    Polynomial Long Division
Dividing :  x⁵ - x⁴ + x³ - 1 
                              ("Dividend")
By         :    x - 1    ("Divisor")

Quotient :  x⁴+x²+x+1  Remainder:  0 

Polynomial Roots Calculator :

 4.6    Find roots (zeroes) of :       F(x) = x⁴+x²+x+1

     See theory in step 4.4
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

Adding fractions that have a common denominator :

 4.7       Adding up the two equivalent fractions

 (x4+x2+x+1) • (x-1) - (x2)     x5 - x4 + x3 - x2 - 1 
 ——————————————————————————  =  —————————————————————
             x2                          x2          

Polynomial Roots Calculator :

 4.8    Find roots (zeroes) of :       F(x) = x⁵ - x⁴ + x³ - x² - 1

     See theory in step 4.4
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 :

  x5 - x4 + x3 - x2 - 1 
  —————————————————————
           x2