Solution - Simplification or other simple results

Step  1  :

Equation at the end of step  1  :

  ((((4•(x5))-(7•(x3)))-(9•(x2)))+18)-((((14•(x5))-(8•(x4)))+11x2)+x)
 Step  2  :

Equation at the end of step  2  :

  ((((4•(x5))-(7•(x3)))-(9•(x2)))+18)-((((14•(x5))-23x4)+11x2)+x)
 Step  3  :

Equation at the end of step  3  :

  ((((4•(x5))-(7•(x3)))-(9•(x2)))+18)-((((2•7x5)-23x4)+11x2)+x)
 Step  4  :

Equation at the end of step  4  :

  ((((4•(x5))-(7•(x3)))-32x2)+18)-(14x5-8x4+11x2+x)
 Step  5  :

Equation at the end of step  5  :

  ((((4•(x5))-7x3)-32x2)+18)-(14x5-8x4+11x2+x)
 Step  6  :

Equation at the end of step  6  :

  (((22x5 -  7x3) -  32x2) +  18) -  (14x5 - 8x4 + 11x2 + x)

Step  7  :

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   -10x⁵ + 8x⁴ - 7x³ - 20x² - x + 18  = 

  -1 • (10x⁵ - 8x⁴ + 7x³ + 20x² + x - 18) 

Trying to factor by pulling out :

 8.2      Factoring:  10x⁵ - 8x⁴ + 7x³ + 20x² + x - 18 

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

Group 1:  7x³ + 20x² 
Group 2:  10x⁵ - 8x⁴ 
Group 3:  x - 18 

Pull out from each group separately :

Group 1:   (7x + 20) • (x²)
Group 2:   (5x - 4) • (2x⁴)
Group 3:   (x - 18) • (1)


Looking for common sub-expressions :

Group 1:   (7x + 20) • (x²)
Group 3:   (x - 18) • (1)
Group 2:   (5x - 4) • (2x⁴)

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 :

 8.3    Find roots (zeroes) of :       F(x) = 10x⁵ - 8x⁴ + 7x³ + 20x² + x - 18
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  10  and the Trailing Constant is  -18.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,5 ,10
 of the Trailing Constant :  1 ,2 ,3 ,6 ,9 ,18

 Let us test ....

Note - For tidiness, printing of 31 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

  -10x5 + 8x4 - 7x3 - 20x2 - x + 18