Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  
 Step  2  :

Equation at the end of step  2  :

  
 Step  3  :

Equation at the end of step  3  :

Step  4  :

            12x4 - 8x3 + 5
 Simplify   ——————————————
              ( -22•3x4)  

Polynomial Roots Calculator :

 4.1    Find roots (zeroes) of :       F(x) = 12x⁴ - 8x³ + 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  12  and the Trailing Constant is  5.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,3 ,4 ,6 ,12
 of the Trailing Constant :  1 ,5

 Let us test ....

Note - For tidiness, printing of 19 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Polynomial Long Division :

 4.2    Polynomial Long Division
Dividing :  12x⁴ - 8x³ + 5 
                              ("Dividend")
By         :     -2² • 3x⁴    ("Divisor")

Quotient :  1 
Remainder :  -8x³ + 5 

Final result :

  12x4 - 8x3 + 5
  ——————————————
    ( -22•3x4)