Solution - Finding the roots of polynomials

Step  1  :

Equation at the end of step  1  :

Step  2  :

            14x3 + x2 - 30x + 13
 Simplify   ————————————————————
                  7x - 10       

Checking for a perfect cube :

 2.1    14x³ + x² - 30x + 13  is not a perfect cube

Trying to factor by pulling out :

 2.2      Factoring:  14x³ + x² - 30x + 13 

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1:  -30x + 13 
Group 2:  14x³ + x² 

Pull out from each group separately :

Group 1:   (-30x + 13) • (1) = (30x - 13) • (-1)
Group 2:   (14x + 1) • (x²)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

 2.3    Find roots (zeroes) of :       F(x) = 14x³ + x² - 30x + 13
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  14  and the Trailing Constant is  13.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,7 ,14
 of the Trailing Constant :  1 ,13

 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
   14x³ + x² - 30x + 13 
can be divided with  2x - 1 

Polynomial Long Division :

 2.4    Polynomial Long Division
Dividing :  14x³ + x² - 30x + 13 
                              ("Dividend")
By         :    2x - 1    ("Divisor")

Quotient :  7x²+4x-13  Remainder:  0 

Trying to factor by splitting the middle term

 2.5     Factoring  7x²+4x-13 

The first term is,  7x²  its coefficient is  7 .
The middle term is,  +4x  its coefficient is  4 .
The last term, "the constant", is  -13 

Step-1 : Multiply the coefficient of the first term by the constant   7 • -13 = -91 

Step-2 : Find two factors of  -91  whose sum equals the coefficient of the middle term, which is   4 .

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

Final result :

  (7x2 + 4x - 13) • (2x - 1)
  ——————————————————————————
           7x - 10