Solution - Polynomial long division

Reformatting the input :

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  :

  ((((x4)-(6•(x3)))+13x2)-24x)+36
 Step  2  :

Equation at the end of step  2  :

  ((((x4) -  (2•3x3)) +  13x2) -  24x) +  36

Step  3  :

Polynomial Roots Calculator :

 3.1    Find roots (zeroes) of :       F(x) = x⁴-6x³+13x²-24x+36
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  36.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,9 ,12 ,18 ,36

 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
   x⁴-6x³+13x²-24x+36 
can be divided with  x-3 

Polynomial Long Division :

 3.2    Polynomial Long Division
Dividing :  x⁴-6x³+13x²-24x+36 
                              ("Dividend")
By         :    x-3    ("Divisor")

Quotient :  x³-3x²+4x-12  Remainder:  0 

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = x³-3x²+4x-12

     See theory in step 3.1
In this case, the Leading Coefficient is  1  and the Trailing Constant is  -12.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,3 ,4 ,6 ,12

 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
   x³-3x²+4x-12 
can be divided with  x-3 

Polynomial Long Division :

 3.4    Polynomial Long Division
Dividing :  x³-3x²+4x-12 
                              ("Dividend")
By         :    x-3    ("Divisor")

Quotient :  x²+4  Remainder:  0 

Polynomial Roots Calculator :

 3.5    Find roots (zeroes) of :       F(x) = x²+4

     See theory in step 3.1
In this case, the Leading Coefficient is  1  and the Trailing Constant is  4.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Multiplying Exponential Expressions :

 3.6    Multiply  (x-3)  by  (x-3) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x-3)  and the exponents are :
          1 , as  (x-3)  is the same number as  (x-3)¹ 
 and   1 , as  (x-3)  is the same number as  (x-3)¹ 
The product is therefore,  (x-3)⁽¹⁺¹⁾ = (x-3)² 

Final result :

  (x2 + 4) • (x - 3)2