Solution - Reducing fractions to their lowest terms

Step  1  :

             1
 Simplify   ——
            x2

Equation at the end of step  1  :

                  1
  ((x2) +  2) +  ——
                 x2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

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

               x2 + 2     (x2 + 2) • x2
     x2 + 2 =  ——————  =  —————————————
                 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

Polynomial Roots Calculator :

 2.2    Find roots (zeroes) of :       F(x) = x² + 2
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  2.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2

 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:

 (x2+2) • x2 + 1     x4 + 2x2 + 1
 ———————————————  =  ————————————
       x2                 x2     

Trying to factor by splitting the middle term

 2.4     Factoring  x⁴ + 2x² + 1 

The first term is,  x⁴  its coefficient is  1 .
The middle term is,  +2x²  its coefficient is  2 .
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   2 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  1  and  1 
                     x⁴ + 1x² + 1x² + 1

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x² • (x²+1)
              Add up the last 2 terms, pulling out common factors :
                     1 • (x²+1)
Step-5 : Add up the four terms of step 4 :
                    (x²+1)  •  (x²+1)
             Which is the desired factorization

Polynomial Roots Calculator :

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

     See theory in step 2.2
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

Polynomial Roots Calculator :

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

     See theory in step 2.2
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

Multiplying Exponential Expressions :

 2.7    Multiply  (x²+1)  by  (x²+1) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x²+1)  and the exponents are :
          1 , as  (x²+1)  is the same number as  (x²+1)¹ 
 and   1 , as  (x²+1)  is the same number as  (x²+1)¹ 
The product is therefore,  (x²+1)⁽¹⁺¹⁾ = (x²+1)² 

Final result :

  (x2 + 1)2
  —————————
     x2