Solution - Finding the roots of polynomials

Step  1  :

Equation at the end of step  1  :

Step  2  :

            x3 - 4x2 + 3
 Simplify   ————————————
                 x      

Polynomial Roots Calculator :

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

 The factor(s) are:

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

 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 - 4x² + 3 
can be divided with  x - 1 

Polynomial Long Division :

 2.2    Polynomial Long Division
Dividing :  x³ - 4x² + 3 
                              ("Dividend")
By         :    x - 1    ("Divisor")

Quotient :  x²-3x-3  Remainder:  0 

Trying to factor by splitting the middle term

 2.3     Factoring  x²-3x-3 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -3x  its coefficient is  -3 .
The last term, "the constant", is  -3 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -3 = -3 

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

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Polynomial Long Division :

 2.4    Polynomial Long Division
Dividing :  x²-3x-3 
                              ("Dividend")
By         :    x    ("Divisor")

Quotient :  x-3 
Remainder :  -3 

Final result :

  (x2 - 3x - 3) • (x - 1)
  ———————————————————————
             x