Question

Factorize: 6x3-5x2-13x+12

Answer 1

 6x3-5x2-13x+12=0 

Three solutions were found :

 x = 4/3 = 1.333

 x = -3/2 = -1.500

 x = 1

Step by step solution :

Step  1  :

Equation at the end of step  1  :

(((6 • (x3)) - 5x2) - 13x) + 12 = 0

Step  2  :

Equation at the end of step  2  :

(((2•3x3) - 5x2) - 13x) + 12 = 0

Step  3  :

Checking for a perfect cube :

 3.1    6x3-5x2-13x+12  is not a perfect cube 

Trying to factor by pulling out :

 3.2      Factoring:  6x3-5x2-13x+12 

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

Group 1:  -13x+12 
Group 2:  6x3-5x2 

Pull out from each group separately :

Group 1:   (-13x+12) • (1) = (13x-12) • (-1)
Group 2:   (6x-5) • (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 :

 3.3    Find roots (zeroes) of :       F(x) = 6x3-5x2-13x+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  6  and the Trailing Constant is  12. 

 The factor(s) are: 

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

 Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      14.00        -1     2      -0.50      16.50        -1     3      -0.33      15.56        -1     6      -0.17      14.00        -2     1      -2.00      -30.00        -2     3      -0.67      16.67        -3     1      -3.00      -156.00        -3     2      -1.50      0.00    2x+3      -4     1      -4.00      -400.00        -4     3      -1.33      6.22        -6     1      -6.00     -1386.00        -12     1     -12.00     -10920.00        1     1      1.00      0.00    x-1      1     2      0.50      5.00        1     3      0.33      7.33        1     6      0.17      9.72        2     1      2.00      14.00        2     3      0.67      2.89        3     1      3.00      90.00        3     2      1.50      1.50        4     1      4.00      264.00        4     3      1.33      0.00    3x-4      6     1      6.00      1050.00        12     1      12.00      9504.00   


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 
   6x3-5x2-13x+12 
can be divided by 3 different polynomials,including by  3x-4 

Polynomial Long Division :

 3.4    Polynomial Long Division 
Dividing :  6x3-5x2-13x+12 
                              ("Dividend")
By         :    3x-4    ("Divisor")

dividend  6x3 - 5x2 - 13x + 12 - divisor * 2x2   6x3 - 8x2     remainder    3x2 - 13x + 12 - divisor * x1     3x2 - 4x   remainder    - 9x + 12 - divisor * -3x0     - 9x + 12 remainder       0

Quotient :  2x2+x-3  Remainder:  0 

Trying to factor by splitting the middle term

 3.5     Factoring  2x2+x-3 

The first term is,  2x2  its coefficient is  2 .
The middle term is,  +x  its coefficient is  1 .
The last term, "the constant", is  -3 

Step-1 : Multiply the coefficient of the first term by the constant   2 • -3 = -6 

Step-2 : Find two factors of  -6  whose sum equals the coefficient of the middle term, which is   1 .

     -6   +   1   =   -5     -3   +   2   =   -1     -2   +   3   =   1   That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -2  and  3 
                     2x2 - 2x + 3x - 3

Step-4 : Add up the first 2 terms, pulling out like factors :
                    2x • (x-1)
              Add up the last 2 terms, pulling out common factors :
                    3 • (x-1)
Step-5 : Add up the four terms of step 4 :
                    (2x+3)  •  (x-1)
             Which is the desired factorization

Equation at the end of step  3  :

(x - 1) • (2x + 3) • (3x - 4) =

Answer 2

hope it helps you.....