(2x^4 + 3x^3 - 2x^2 - 3x -6) (2x + 3)

Step 1 :Equation at the end of step 1

 2•(x4))+(3•(x3)))-2x2)-3x)-6)•(2x+3)

Step 2 :

Equation at the end of step 2 :

 2•(x4))+3x3)-2x2)-3x)-6)•(2x+3)

Step 3 :

Equation at the end of step 3 :

 2x4 +  3x3) -  2x2) -  3x) -  6) • (2x + 3)

Step 4 :

Polynomial Roots Calculator :

4.1    Find roots (zeroes) of :       F(x) = 2x4+3x3-2x2-3x-6

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  -6.

The factor(s) are:

of the Leading Coefficient :  1,2

of the Trailing Constant :  1 ,2 ,3 ,6

Let us test

 P    Q    P/Q    F(P/Q)    Divisor

     -1       1        -1.00        -6.00    

     -1       2        -0.50        -5.25    

     -2       1        -2.00        0.00      x+2

     -3       1        -3.00        66.00    

     -3       2        -1.50        -6.00    

     -6       1        -6.00        1884.00    

     1       1        1.00        -6.00    

     1       2        0.50        -7.50    

     2       1        2.00        36.00    

     3       1        3.00        210.00    

     3       2        1.50        5.25    

     6       1        6.00        3144.00    

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+3x3-2x2-3x-6

can be divided with  x+2

Polynomial Long Division :

4.2    Polynomial Long Division

Dividing :  2x4+3x3-2x2-3x-6

                             ("Dividend")

By         :    x+2    ("Divisor")

dividend     2x4  +  3x3  -  2x2  -  3x  -  6

- divisor  * 2x3     2x4  +  4x3            

remainder      -  x3  -  2x2  -  3x  -  6

- divisor  * -x2      -  x3  -  2x2        

remainder              -  3x  -  6

- divisor  * 0x1                    

remainder              -  3x  -  6

- divisor  * -3x0              -  3x  -  6

remainder                    0

Quotient :  2x3-x2-3  Remainder:  0

Polynomial Roots Calculator :

4.3    Find roots (zeroes) of :       F(x) = 2x3-x2-3

    See theory in step 4.1

In this case, the Leading Coefficient is  2  and the Trailing Constant is  -3.

The factor(s) are:

of the Leading Coefficient :  1,2

of the Trailing Constant :  1 ,3

Let us test

 P    Q    P/Q    F(P/Q)    Divisor

     -1       1        -1.00        -6.00    

     -1       2        -0.50        -3.50    

     -3       1        -3.00        -66.00    

     -3       2        -1.50        -12.00    

     1       1        1.00        -2.00    

     1       2        0.50        -3.00    

     3       1        3.00        42.00    

     3       2        1.50        1.50    

Polynomial Roots Calculator found no rational roots

Final result :

 (2x3 - x2 - 3) • (x + 2) • (2x + 3)