Solution - Reducing fractions to their lowest terms

Step  1  :

             11
 Simplify   ———
            405

Equation at the end of step  1  :

           11
  (f4) -  ———
          405
 Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  405  as the denominator :

           f4     f4 • 405
     f4 =  ——  =  ————————
           1        405   

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:

 f4 • 405 - (11)     405f4 - 11
 ———————————————  =  ——————————
       405              405    

Trying to factor as a Difference of Squares :

 2.3      Factoring:  405f⁴ - 11 

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 :  405  is not a square !!

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

Polynomial Roots Calculator :

 2.4    Find roots (zeroes) of :       F(f) = 405f⁴ - 11
Polynomial Roots Calculator is a set of methods aimed at finding values of  f  for which   F(f)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  f  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  405  and the Trailing Constant is  -11.

 The factor(s) are:

of the Leading Coefficient :  1,3 ,5 ,9 ,15 ,27 ,45 ,81 ,135 ,405
 of the Trailing Constant :  1 ,11

 Let us test ....

Note - For tidiness, printing of 35 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  405f4 + 11
  ——————————
     405