Solution - Reducing fractions to their lowest terms

Step  1  :

            5
 Simplify   —
            x

Equation at the end of step  1  :

         5     
  (5x +  —) -  1
         x     

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

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

           5x     5x • x
     5x =  ——  =  ——————
           1        x   

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:

 5x • x + 5     5x2 + 5
 ——————————  =  ———————
     x             x   

Equation at the end of step  2  :

  (5x2 + 5)    
  ————————— -  1
      x        

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

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

         1     1 • x
    1 =  —  =  —————
         1       x  

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   5x² + 5  =   5 • (x² + 1) 

Polynomial Roots Calculator :

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

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 5 • (x2+1) - (x)     5x2 - x + 5
 ————————————————  =  ———————————
        x                  x     

Trying to factor by splitting the middle term

 4.4     Factoring  5x² - x + 5 

The first term is,  5x²  its coefficient is  5 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is  +5 

Step-1 : Multiply the coefficient of the first term by the constant   5 • 5 = 25 

Step-2 : Find two factors of  25  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

Final result :

  5x2 - x + 5
  ———————————
       x