Solution - Finding the roots of polynomials

Step  1  :

             1
 Simplify   ——
            a3

Equation at the end of step  1  :

            1     
  ((a2) +  ——) +  1
           a3     
 Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  a³  as the denominator :

           a2     a2 • a3
     a2 =  ——  =  ———————
           1        a3   

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

Adding fractions that have a common denominator :

 2.2       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:

 a2 • a3 + 1     a5 + 1
 ———————————  =  ——————
     a3            a3  

Equation at the end of step  2  :

  (a5 + 1)    
  ———————— +  1
     a3       

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  a³  as the denominator :

         1     1 • a3
    1 =  —  =  ——————
         1       a3  

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(a) = a⁵ + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  a  for which   F(a)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  a  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
   a⁵ + 1 
can be divided with  a + 1 

Polynomial Long Division :

 3.3    Polynomial Long Division
Dividing :  a⁵ + 1 
                              ("Dividend")
By         :    a + 1    ("Divisor")

Quotient :  a⁴-a³+a²-a+1  Remainder:  0 

Polynomial Roots Calculator :

 3.4    Find roots (zeroes) of :       F(a) = a⁴-a³+a²-a+1

     See theory in step 3.2
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 :

 3.5       Adding up the two equivalent fractions

 (a4-a3+a2-a+1) • (a+1) + a3      a5 + a3 + 1
 ———————————————————————————  =  ———————————
             a3                      a3     

Polynomial Roots Calculator :

 3.6    Find roots (zeroes) of :       F(a) = a⁵ + a³ + 1

     See theory in step 3.2
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 :

  a5 + a3 + 1
  ———————————
      a3