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 Â Â Find roots (zeroes) of : Â Â Â 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 Â
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 : Â 1,2
of the Trailing Constant : Â 1 ,2 ,3 ,6
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     -6.00  Â
   -1    2     -0.50     -5.25  Â
   -2    1     -2.00     0.00    x+2
   -3    1     -3.00     66.00  Â
   -3    2     -1.50     -6.00  Â
   -6    1     -6.00     1884.00  Â
   1    1     1.00     -6.00  Â
   1    2     0.50     -7.50  Â
   2    1     2.00     36.00  Â
   3    1     3.00     210.00  Â
   3    2     1.50     5.25  Â
   6    1     6.00     3144.00  Â
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 Â Â Polynomial Long Division
Dividing : Â 2x4+3x3-2x2-3x-6
               ("Dividend")
By     :   x+2   ("Divisor")
dividend   2x4  +  3x3  -  2x2  -  3x  -  6
- divisor  * 2x3   2x4  +  4x3      Â
remainder    -  x3  -  2x2  -  3x  -  6
- divisor  * -x2    -  x3  -  2x2    Â
remainder        -  3x  -  6
- divisor  * 0x1          Â
remainder        -  3x  -  6
- divisor  * -3x0        -  3x  -  6
remainder           0
Quotient : Â 2x3-x2-3 Â Remainder: Â 0
Polynomial Roots Calculator :
4.3 Â Â Find roots (zeroes) of : Â Â Â 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 : Â 1,2
of the Trailing Constant : Â 1 ,3
Let us test
 P   Q   P/Q   F(P/Q)   Divisor
   -1    1     -1.00     -6.00  Â
   -1    2     -0.50     -3.50  Â
   -3    1     -3.00     -66.00  Â
   -3    2     -1.50     -12.00  Â
   1    1     1.00     -2.00  Â
   1    2     0.50     -3.00  Â
   3    1     3.00     42.00  Â
   3    2     1.50     1.50  Â
Polynomial Roots Calculator found no rational roots
Final result :
 (2x3 - x2 - 3) • (x + 2) • (2x + 3)
