Solution - Reducing fractions to their lowest terms

Step  1  :

            m
 Simplify   —
            3

Equation at the end of step  1  :

         m          
  (5 -  (— • m)) -  15
         3          
 Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  3  as the denominator :

          5     5 • 3
     5 =  —  =  —————
          1       3  

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:

 5 • 3 - (m2)     15 - m2
 ————————————  =  ———————
      3              3   

Equation at the end of step  2  :

  (15 - m2)    
  ————————— -  15
      3        

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  3  as the denominator :

          15     15 • 3
    15 =  ——  =  ——————
          1        3   

Trying to factor as a Difference of Squares :

 3.2      Factoring:  15 - m² 

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 :  15  is not a square !!

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

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

 (15-m2) - (15 • 3)     -m2 - 30
 ——————————————————  =  ————————
         3                 3    

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   -m² - 30  =   -1 • (m² + 30) 

Polynomial Roots Calculator :

 4.2    Find roots (zeroes) of :       F(m) = m² + 30
Polynomial Roots Calculator is a set of methods aimed at finding values of  m  for which   F(m)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  m  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  30.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,5 ,6 ,10 ,15 ,30

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  +m2 + 30
  ————————
     3