Solution - Finding the roots of polynomials

Step  1  :

Equation at the end of step  1  :

  ((3x2 • (x + 3)) • (x - 16)) +  (x2 - x - 12)

Step  2  :

Equation at the end of step  2  :

  (3x2 • (x + 3) • (x - 16)) +  (x2 - x - 12)

Step  3  :

Equation at the end of step  3  :

  3x2 • (x + 3) • (x - 16) +  (x2 - x - 12)

Step  4  :

Polynomial Roots Calculator :

 4.1    Find roots (zeroes) of :       F(x) = 3x⁴-39x³-143x²-x-12
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  3  and the Trailing Constant is  -12.

 The factor(s) are:

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

 Let us test ....

Note - For tidiness, printing of 13 checks which found no root was suppressed

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
   3x⁴-39x³-143x²-x-12 
can be divided with  x+3 

Polynomial Long Division :

 4.2    Polynomial Long Division
Dividing :  3x⁴-39x³-143x²-x-12 
                              ("Dividend")
By         :    x+3    ("Divisor")

Quotient :  3x³-48x²+x-4  Remainder:  0 

Polynomial Roots Calculator :

 4.3    Find roots (zeroes) of :       F(x) = 3x³-48x²+x-4

     See theory in step 4.1
In this case, the Leading Coefficient is  3  and the Trailing Constant is  -4.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Final result :

  (3x3 - 48x2 + x - 4) • (x + 3)