Solution - Reducing fractions to their lowest terms

Step  1  :

             6
 Simplify   ——
            a2

Equation at the end of step  1  :

                  6
  ((a2) +  1) -  ——
                 a2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

               a2 + 1     (a2 + 1) • a2
     a2 + 1 =  ——————  =  —————————————
                 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

Polynomial Roots Calculator :

 2.2    Find roots (zeroes) of :       F(a) = 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

Adding fractions that have a common denominator :

 2.3       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:

 (a2+1) • a2 - (6)     a4 + a2 - 6
 —————————————————  =  ———————————
        a2                 a2     

Trying to factor by splitting the middle term

 2.4     Factoring  a⁴ + a² - 6 

The first term is,  a⁴  its coefficient is  1 .
The middle term is,  +a²  its coefficient is  1 .
The last term, "the constant", is  -6 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -6 = -6 

Step-2 : Find two factors of  -6  whose sum equals the coefficient of the middle term, which is   1 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -2  and  3 
                     a⁴ - 2a² + 3a² - 6

Step-4 : Add up the first 2 terms, pulling out like factors :
                    a² • (a²-2)
              Add up the last 2 terms, pulling out common factors :
                    3 • (a²-2)
Step-5 : Add up the four terms of step 4 :
                    (a²+3)  •  (a²-2)
             Which is the desired factorization

Polynomial Roots Calculator :

 2.5    Find roots (zeroes) of :       F(a) = a²+3

     See theory in step 2.2
In this case, the Leading Coefficient is  1  and the Trailing Constant is  3.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 2.6      Factoring:  a²-2 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 2 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Final result :

  (a2 + 3) • (a2 - 2)
  ———————————————————
          a2