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2015-03-02T21:44:37+05:30

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3 x + 2 is factor of  P(x)  = 6 x³ + 7 x² - x - 2  ??

   If  3 (x + 2/3)  is a factor of  the expression, P(x = - 2/3)  should be 0.
       Reminder of division of P(x) by (x+2/3) is  P( - 2/3)
     P(2/3) = -48/27  + 28/9 + 2/3 - 2 = 0
As the reminder is  zero,  (x + 2/3) is a factor. 
 
 Hence,  6 x³ + 7 x² - x - 2 = 3 (x + 2/3)  ( a x² + b x + c) +  0

 comparing coefficients of x³ on both sides, 3 a = 6  => a = 2
     coefficients of the constant terms:    -2 = 2 c    => c = -1
           finding coefficient of x² term:  7 = 2 a + 3 b    =>  b = 1

             quotient :  2 x² + x - 1
==========================================

3 x + 2  )  6 x³  + 7 x² - x - 2  (  2 x²  + x  - 1
               6 x³  + 4 x²
           ============
                       3 x² - x
                       3 x² + 2 x
                     ===========
                               - 3 x - 2
                               - 3 x - 2
                             =======
                               0

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