Solution - Factoring binomials using the difference of squares

Step  1  :

             1
 Simplify   ——
            a2

Equation at the end of step  1  :

         1     
  (a +  ——) +  1
        a2     

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  a²  as the denominator :

          a     a • a2
     a =  —  =  ——————
          1       a2  

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:

 a • a2 + 1     a3 + 1
 ——————————  =  ——————
     a2           a2  

Equation at the end of step  2  :

  (a3 + 1)    
  ———————— +  1
     a2       

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  a²  as the denominator :

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

Trying to factor as a Sum of Cubes :

 3.2      Factoring:  a³ + 1 

Theory : A sum of two perfect cubes,  a³ + b³ can be factored into  :
             (a+b) • (a²-ab+b²)
Proof  : (a+b) • (a²-ab+b²) =
    a³-a²b+ab²+ba²-b²a+b³ =
    a³+(a²b-ba²)+(ab²-b²a)+b³=
    a³+0+0+b³=
    a³+b³

Check :  1  is the cube of   1 
Check :  a³ is the cube of   a¹

Factorization is :
             (a + 1)  •  (a² - a + 1) 

Trying to factor by splitting the middle term

 3.3     Factoring  a² - a + 1 

The first term is,  a²  its coefficient is  1 .
The middle term is,  -a  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 :

 3.4       Adding up the two equivalent fractions

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

Polynomial Roots Calculator :

 3.5    Find roots (zeroes) of :       F(a) = a³ + a² + 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 ....

Polynomial Roots Calculator found no rational roots

Final result :

  a3 + a2 + 1
  ———————————
      a2