Solution - Finding the roots of polynomials

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "p2"   was replaced by   "p^2".  2 more similar replacement(s).

Step  1  :

Equation at the end of step  1  :

  ((((2•(p4))-(p3))-22p2)-8p)+8
 Step  2  :

Equation at the end of step  2  :

  (((2p4 -  p3) -  22p2) -  8p) +  8

Step  3  :

Polynomial Roots Calculator :

 3.1    Find roots (zeroes) of :       F(p) = 2p⁴-p³-4p²-8p+8
Polynomial Roots Calculator is a set of methods aimed at finding values of  p  for which   F(p)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  p  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  2  and the Trailing Constant is  8.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,2 ,4 ,8

 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
   2p⁴-p³-4p²-8p+8 
can be divided with  p-2 

Polynomial Long Division :

 3.2    Polynomial Long Division
Dividing :  2p⁴-p³-4p²-8p+8 
                              ("Dividend")
By         :    p-2    ("Divisor")

Quotient :  2p³+3p²+2p-4  Remainder:  0 

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(p) = 2p³+3p²+2p-4

     See theory in step 3.1
In this case, the Leading Coefficient is  2  and the Trailing Constant is  -4.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1 ,2 ,4

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  (2p3 + 3p2 + 2p - 4) • (p - 2)