Solution - Finding the roots of polynomials

Step  1  :

Equation at the end of step  1  :

  ((((5•(x4))-(8•(x3)))-(2•3x2))+8) 
  ————————————————————————————————— ÷ (x+1)
                (x-2)              
 Step  2  :

Equation at the end of step  2  :

  ((((5•(x4))-23x3)-(2•3x2))+8) 
  ————————————————————————————— ÷ (x+1)
              (x-2)            
 Step  3  :

Equation at the end of step  3  :

  (((5x4-23x3)-(2•3x2))+8) 
  ———————————————————————— ÷ (x+1)
           (x-2)          

Step  4  :

            5x4 - 8x3 - 6x2 + 8 
 Simplify   ———————————————————
                   x - 2       

Checking for a perfect cube :

 4.1    5x⁴ - 8x³ - 6x² + 8  is not a perfect cube

Trying to factor by pulling out :

 4.2      Factoring:  5x⁴ - 8x³ - 6x² + 8 

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

Group 1:  -6x² + 8 
Group 2:  -8x³ + 5x⁴ 

Pull out from each group separately :

Group 1:   (3x² - 4) • (-2)
Group 2:   (5x - 8) • (x³)

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 :

 4.3    Find roots (zeroes) of :       F(x) = 5x⁴ - 8x³ - 6x² + 8
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  5  and the Trailing Constant is  8.

 The factor(s) are:

of the Leading Coefficient :  1,5
 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
   5x⁴ - 8x³ - 6x² + 8 
can be divided with  x - 2 

Polynomial Long Division :

 4.4    Polynomial Long Division
Dividing :  5x⁴ - 8x³ - 6x² + 8 
                              ("Dividend")
By         :    x - 2    ("Divisor")

Quotient :  5x³+2x²-2x-4  Remainder:  0 

Polynomial Roots Calculator :

 4.5    Find roots (zeroes) of :       F(x) = 5x³+2x²-2x-4

     See theory in step 4.3
In this case, the Leading Coefficient is  5  and the Trailing Constant is  -4.

 The factor(s) are:

of the Leading Coefficient :  1,5
 of the Trailing Constant :  1 ,2 ,4

 Let us test ....

Polynomial Roots Calculator found no rational roots

Canceling Out :

 4.6    Cancel out  (x-2)  which appears on both sides of the fraction line.

Equation at the end of step  4  :

  (5x3 + 2x2 - 2x - 4)
  ————————————————————
        (x + 1)       

Step  5  :

            5x3 + 2x2 - 2x - 4
 Simplify   ——————————————————
                  x + 1       

Polynomial Long Division :

 5.1    Polynomial Long Division
Dividing :  5x³ + 2x² - 2x - 4 
                              ("Dividend")
By         :    x + 1    ("Divisor")

Quotient :  5x² - 3x + 1 
Remainder :  -5 

Final result :

  5x3 + 2x2 - 2x - 4
  ——————————————————
        x + 1