Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  (((0-(g4))-(5•(g3)))-(10•(g2)))+(((9•(g3))+3g2)-1)
 Step  2  :

Equation at the end of step  2  :

  (((0-(g4))-(5•(g3)))-(10•(g2)))+((32g3+3g2)-1)
 Step  3  :

Equation at the end of step  3  :

  (((0-(g4))-(5•(g3)))-(2•5g2))+(9g3+3g2-1)
 Step  4  :

Equation at the end of step  4  :

  (((0-(g4))-5g3)-(2•5g2))+(9g3+3g2-1)

Step  5  :

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   -g⁴ + 4g³ - 7g² - 1  = 

  -1 • (g⁴ - 4g³ + 7g² + 1) 

Checking for a perfect cube :

 6.2    g⁴ - 4g³ + 7g² + 1  is not a perfect cube

Trying to factor by pulling out :

 6.3      Factoring:  g⁴ - 4g³ + 7g² + 1 

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

Group 1:  7g² + 1 
Group 2:  g⁴ - 4g³ 

Pull out from each group separately :

Group 1:   (7g² + 1) • (1)
Group 2:   (g - 4) • (g³)

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 :

 6.4    Find roots (zeroes) of :       F(g) = g⁴ - 4g³ + 7g² + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  g  for which   F(g)=0  

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

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  -g4 + 4g3 - 7g2 - 1