Solution - Finding the roots of polynomials

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   x²¹ - x  =   x • (x²⁰ - 1) 

Trying to factor as a Difference of Squares :

 2.2      Factoring:  x²⁰ - 1 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1
Check :  x²⁰  is the square of  x¹⁰ 

Factorization is :       (x¹⁰ + 1)  •  (x¹⁰ - 1) 

Trying to factor as a Difference of Squares :

 2.3      Factoring:  x¹⁰ - 1 

Check : 1 is the square of 1
Check :  x¹⁰  is the square of  x⁵ 

Factorization is :       (x⁵ + 1)  •  (x⁵ - 1) 

Polynomial Roots Calculator :

 2.4    Find roots (zeroes) of :       F(x) = x⁵ + 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  1  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1
 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
   x⁵ + 1 
can be divided with  x + 1 

Polynomial Long Division :

 2.5    Polynomial Long Division
Dividing :  x⁵ + 1 
                              ("Dividend")
By         :    x + 1    ("Divisor")

Quotient :  x⁴-x³+x²-x+1  Remainder:  0 

Polynomial Roots Calculator :

 2.6    Find roots (zeroes) of :       F(x) = x⁴-x³+x²-x+1

     See theory in step 2.4
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

Polynomial Roots Calculator :

 2.7    Find roots (zeroes) of :       F(x) = x⁵-1

     See theory in step 2.4
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 ....

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
   x⁵-1 
can be divided with  x-1 

Polynomial Long Division :

 2.8    Polynomial Long Division
Dividing :  x⁵-1 
                              ("Dividend")
By         :    x-1    ("Divisor")

Quotient :  x⁴+x³+x²+x+1  Remainder:  0 

Polynomial Roots Calculator :

 2.9    Find roots (zeroes) of :       F(x) = x⁴+x³+x²+x+1

     See theory in step 2.4
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 :

  x•(x10+1)•(x4-x3+x2-x+1)•(x+1)•(x4+x3+x2+x+1)•(x-1)