Solution - Reducing fractions to their lowest terms

Step  1  :

             s
 Simplify   ——
            s2

Dividing exponential expressions :

 1.1    s¹ divided by s² = s^{(1 - 2)} = s^(-1) = ¹/_s¹ = ¹/_s

Equation at the end of step  1  :

        1           
  l • ((— +  4s) +  13)
        s           

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

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

          4s     4s • s
    4s =  ——  =  ——————
          1        s   

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:

 1 + 4s • s     4s2 + 1
 ——————————  =  ———————
     s             s   

Equation at the end of step  2  :

       (4s2 + 1)    
  l • (————————— +  13)
           s        

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

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

          13     13 • s
    13 =  ——  =  ——————
          1        s   

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(s) = 4s² + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  s  for which   F(s)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  s  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  4  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

 (4s2+1) + 13 • s     4s2 + 13s + 1
 ————————————————  =  —————————————
        s                   s      

Equation at the end of step  3  :

      (4s2 + 13s + 1)
  l • ———————————————
             s       

Step  4  :

Trying to factor by splitting the middle term

 4.1     Factoring  4s²+13s+1 

The first term is,  4s²  its coefficient is  4 .
The middle term is,  +13s  its coefficient is  13 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 1 = 4 

Step-2 : Find two factors of  4  whose sum equals the coefficient of the middle term, which is   13 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  l • (4s2 + 13s + 1)
  ———————————————————
           s