Solution - Reducing fractions to their lowest terms

Step  1  :

            5
 Simplify   —
            x

Equation at the end of step  1  :

                         5     
  (((4 • (x2)) -  3x) -  —) -  2
                         x     
 Step  2  :

Equation at the end of step  2  :

                   5     
  ((22x2 -  3x) -  —) -  2
                   x     

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

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

                 4x2 - 3x     (4x2 - 3x) • x
     4x2 - 3x =  ————————  =  ——————————————
                    1               x       

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

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   4x² - 3x  =   x • (4x - 3) 

Adding fractions that have a common denominator :

 4.2       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:

 x • (4x-3) • x - (5)     4x3 - 3x2 - 5
 ————————————————————  =  —————————————
          x                     x      

Equation at the end of step  4  :

  (4x3 - 3x2 - 5)    
  ——————————————— -  2
         x           

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

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

         2     2 • x
    2 =  —  =  —————
         1       x  

Polynomial Roots Calculator :

 5.2    Find roots (zeroes) of :       F(x) = 4x³ - 3x² - 5
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  4  and the Trailing Constant is  -5.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,5

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (4x3-3x2-5) - (2 • x)     4x3 - 3x2 - 2x - 5
 —————————————————————  =  ——————————————————
           x                       x         

Checking for a perfect cube :

 5.4    4x³ - 3x² - 2x - 5  is not a perfect cube

Trying to factor by pulling out :

 5.5      Factoring:  4x³ - 3x² - 2x - 5 

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

Group 1:  -2x - 5 
Group 2:  4x³ - 3x² 

Pull out from each group separately :

Group 1:   (2x + 5) • (-1)
Group 2:   (4x - 3) • (x²)

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 :

 5.6    Find roots (zeroes) of :       F(x) = 4x³ - 3x² - 2x - 5

     See theory in step 5.2
In this case, the Leading Coefficient is  4  and the Trailing Constant is  -5.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,5

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  4x3 - 3x2 - 2x - 5
  ——————————————————
          x