Solution - Power equations

Step  1  :

Equation at the end of step  1  :

      d
  (((——•4)•(x5))+(5•(x4)))+3x2
     dx
 Step  2  :

Equation at the end of step  2  :

      d
  (((——•4)•(x5))+5x4)+3x2
     dx
 Step  3  :
             d
 Simplify   ——
            dx

Canceling Out :

 3.1    Canceling out d as it appears on both sides of the fraction line

Equation at the end of step  3  :

     1                         
  (((— • 4) • x5) +  5x4) +  3x2
     x                         

Step  4  :

Dividing exponential expressions :

 4.1    x⁵ divided by x¹ = x^{(5 - 1)} = x⁴

Equation at the end of step  4  :

  (4x4 +  5x4) +  3x2

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   9x⁴ + 3x²  =   3x² • (3x² + 1) 

Polynomial Roots Calculator :

 6.2    Find roots (zeroes) of :       F(x) = 3x² + 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  3  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,3
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  3x2 • (3x2 + 1)