Solution - Finding the roots of polynomials

Step  1  :

Multiplying exponential expressions :

 1.1    x¹ multiplied by x³ = x^{(1 + 3)} = x⁴

Equation at the end of step  1  :

  ((((x2) +  2x4) -  x2) +  x) -  1

Step  2  :

Polynomial Roots Calculator :

 2.1    Find roots (zeroes) of :       F(x) = 2x⁴+x-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  2  and the Trailing Constant is  -1.

 The factor(s) are:

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

 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
   2x⁴+x-1 
can be divided with  x+1 

Polynomial Long Division :

 2.2    Polynomial Long Division
Dividing :  2x⁴+x-1 
                              ("Dividend")
By         :    x+1    ("Divisor")

Quotient :  2x³-2x²+2x-1  Remainder:  0 

Polynomial Roots Calculator :

 2.3    Find roots (zeroes) of :       F(x) = 2x³-2x²+2x-1

     See theory in step 2.1
In this case, the Leading Coefficient is  2  and the Trailing Constant is  -1.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  (2x3 - 2x2 + 2x - 1) • (x + 1)