Solution - Reducing fractions to their lowest terms

Step  1  :

            72
 Simplify   ——
            u2

Equation at the end of step  1  :

                  72             
  (((2 • (u2)) -  ——) +  10u) +  24
                  u2             
 Step  2  :

Equation at the end of step  2  :

           72             
  ((2u2 -  ——) +  10u) +  24
           u2             

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  u²  as the denominator :

            2u2     2u2 • u2
     2u2 =  ———  =  ————————
             1         u2   

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 :

 3.2       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:

 2u2 • u2 - (72)     2u4 - 72
 ———————————————  =  ————————
       u2               u2   

Equation at the end of step  3  :

   (2u4 - 72)            
  (—————————— +  10u) +  24
       u2                

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  u²  as the denominator :

           10u     10u • u2
    10u =  ———  =  ————————
            1         u2   

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   2u⁴ - 72  =   2 • (u⁴ - 36) 

Trying to factor as a Difference of Squares :

 5.2      Factoring:  u⁴ - 36 

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 : 36 is the square of 6
Check :  u⁴  is the square of  u² 

Factorization is :       (u² + 6)  •  (u² - 6) 

Polynomial Roots Calculator :

 5.3    Find roots (zeroes) of :       F(u) = u² + 6
Polynomial Roots Calculator is a set of methods aimed at finding values of  u  for which   F(u)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  u  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  6.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,6

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 5.4      Factoring:  u² - 6 

Check : 6 is not a square !!

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

Adding fractions that have a common denominator :

 5.5       Adding up the two equivalent fractions

 2 • (u2+6) • (u2-6) + 10u • u2     2u4 + 10u3 - 72
 ——————————————————————————————  =  ———————————————
               u2                         u2       

Equation at the end of step  5  :

  (2u4 + 10u3 - 72)    
  ————————————————— +  24
         u2            

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  u²  as the denominator :

          24     24 • u2
    24 =  ——  =  ———————
          1        u2   

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   2u⁴ + 10u³ - 72  =   2 • (u⁴ + 5u³ - 36) 

Polynomial Roots Calculator :

 7.2    Find roots (zeroes) of :       F(u) = u⁴ + 5u³ - 36

     See theory in step 5.3
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -36.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,9 ,12 ,18 ,36

 Let us test ....

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 7.3       Adding up the two equivalent fractions

 2 • (u4+5u3-36) + 24 • u2     2u4 + 10u3 + 24u2 - 72
 —————————————————————————  =  ——————————————————————
            u2                           u2          

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   2u⁴ + 10u³ + 24u² - 72  = 

  2 • (u⁴ + 5u³ + 12u² - 36) 

Checking for a perfect cube :

 8.2    u⁴ + 5u³ + 12u² - 36  is not a perfect cube

Trying to factor by pulling out :

 8.3      Factoring:  u⁴ + 5u³ + 12u² - 36 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  u⁴ + 5u³ 
Group 2:  12u² - 36 

Pull out from each group separately :

Group 1:   (u + 5) • (u³)
Group 2:   (u² - 3) • (12)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 8.4    Find roots (zeroes) of :       F(u) = u⁴ + 5u³ + 12u² - 36

     See theory in step 5.3
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -36.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,9 ,12 ,18 ,36

 Let us test ....

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  2 • (u4 + 5u3 + 12u2 - 36)
  ——————————————————————————
              u2