Solution - Reducing fractions to their lowest terms

Step  1  :

Equation at the end of step  1  :

         3
  (((4•((—•x)-1))•((22•3x2)-3x))•-3x)-(2x-1)
         4

Step  2  :

            3
 Simplify   —
            4

Equation at the end of step  2  :

         3
  (((4•((—•x)-1))•(12x2-3x))•-3x)-(2x-1)
         4

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  4  as the denominator :

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

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:

 3x - (4)     3x - 4
 ————————  =  ——————
    4           4   

Equation at the end of step  3  :

         (3x - 4)                            
  (((4 • ————————) • (12x2 - 3x)) • -3x) -  (2x - 1)
            4                                

Step  4  :

Equation at the end of step  4  :

  (((3x - 4) • (12x2 - 3x)) • -3x) -  (2x - 1)

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   12x² - 3x  =   3x • (4x - 1) 

Equation at the end of step  6  :

  (3x • (3x - 4) • (4x - 1) • -3x) -  (2x - 1)

Step  7  :

Multiplying exponential expressions :

 7.1    x¹ multiplied by x¹ = x^{(1 + 1)} = x²

Equation at the end of step  7  :

  -9x2 • (3x - 4) • (4x - 1) -  (2x - 1)

Step  8  :

Step  9  :

Pulling out like terms :

 9.1     Pull out like factors :

   -108x⁴ + 171x³ - 36x² - 2x + 1  = 

  -1 • (108x⁴ - 171x³ + 36x² + 2x - 1) 

Polynomial Roots Calculator :

 9.2    Find roots (zeroes) of :       F(x) = 108x⁴ - 171x³ + 36x² + 2x - 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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  108  and the Trailing Constant is  -1.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,3 ,4 ,6 ,9 ,12 ,18 ,27 ,36 , etc
 of the Trailing Constant :  1

 Let us test ....

Note - For tidiness, printing of 15 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  +108x4 + 171x3 + 36x2 + 2x + 1