Obtain all other zeroes of the polynomial x4 2x3 13x2 38x 24, if two of its zeroes are 1 and 2

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2".  2 more similar replacement(s).

Step  1  :

Equation at the end of step  1  :

Equation at the end of step  2  :

Step  3  :

Polynomial Roots Calculator :

 3.1    Find roots (zeroes) of :       F(x) = x4+2x3-13x2-38x-24
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 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant : 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

   x4+2x3-13x2-38x-24 

Polynomial Long Division :

 3.2    Polynomial Long Division
Dividing :

 x4+2x3-13x2-38x-24                               ("Dividend")

Quotient :  x3+6x2+11x+6  Remainder:  0 

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = x3+6x2+11x+6

     See theory in step 3.1

of the Leading Coefficient :  1
 of the Trailing Constant : 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

   x3+6x2+11x+6 

Polynomial Long Division :

 3.4    Polynomial Long Division
Dividing :

 x3+6x2+11x+6                               ("Dividend")

Quotient :  x2+3x+2  Remainder:  0 

Trying to factor by splitting the middle term

 3.5     Factoring  x2+3x+2 

The first term is,  x2  its coefficient is  1 .

Step-1 : Multiply the coefficient of the first term by the constant   1 • 2 = 2 

Step-2 : Find two factors of  2  whose sum equals the coefficient of the middle term, which is   3 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  1  and  2 
                     x2 + 1x + 2x + 2Step-4 : Add up the first 2 terms, pulling out like factors :

                    x • (x+1)

                    2 • (x+1)

                    (x+2)  •  (x+1)

Final result :