Solution - Finding the roots of polynomials

Step  1  :

Equation at the end of step  1  :

  (32x4 -  x2) +  2x

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   9x⁴ - x² + 2x  =   x • (9x³ - x + 2) 

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = 9x³ - x + 2
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  9  and the Trailing Constant is  2.

 The factor(s) are:

of the Leading Coefficient :  1,3 ,9
 of the Trailing Constant :  1 ,2

 Let us test ....

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that
   9x³ - x + 2 
can be divided with  3x + 2 

Polynomial Long Division :

 3.3    Polynomial Long Division
Dividing :  9x³ - x + 2 
                              ("Dividend")
By         :    3x + 2    ("Divisor")

Quotient :  3x²-2x+1  Remainder:  0 

Trying to factor by splitting the middle term

 3.4     Factoring  3x²-2x+1 

The first term is,  3x²  its coefficient is  3 .
The middle term is,  -2x  its coefficient is  -2 .
The last term, "the constant", is  +1 

Step-1 : Multiply the coefficient of the first term by the constant   3 • 1 = 3 

Step-2 : Find two factors of  3  whose sum equals the coefficient of the middle term, which is   -2 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Final result :

  x • (3x2 - 2x + 1) • (3x + 2)