Solution - Reducing fractions to their lowest terms

Step  1  :

            q3
 Simplify   ——
            p2

Equation at the end of step  1  :

                q3
  (((p3)•q)-((p•——)•q))-pq2
                p2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

            p3q     p3q • p
     p3q =  ———  =  ———————
             1         p   

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:

 p3q • p - (q4)     p4q - q4
 ——————————————  =  ————————
       p               p    

Equation at the end of step  2  :

  (p4q - q4)    
  —————————— -  pq2
      p         

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

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

           pq2     pq2 • p
    pq2 =  ———  =  ———————
            1         p   

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   p⁴q - q⁴  =   q • (p⁴ - q³) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  p⁴ - q³ 

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 :  p⁴  is the square of  p² 

Check :  q³   is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 q • (p4-q3) - (pq2 • p)      p4q - p2q2 - q4 
 ———————————————————————  =  ———————————————
            p                       p       

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   p⁴q - p²q² - q⁴  =   q • (p⁴ - p²q - q³) 

Trying to factor a multi variable polynomial :

 5.2    Factoring    p⁴ - p²q - q³ 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Final result :

  q • (p4 + p2q + q3) 
  ———————————————————
           p