Factor completely

Question

Factor completely . 2y^3+3y^2−6y−9 Enter the factors as a product of two binomials. Enter the original expression if it cannot be factored.

Answer 1

By the rational Zero Theorem, the possible rational roots are 

±(1, 3, 9, 1/2, 3/2, 9/2).  Testing the various possibilities, we see that -3/2 works.  Divide out by x + 3/2 synthetically:

 

-3/2⌋   2      3      -6       -9

                 -3       0        9                   

         2       0      -6        0

 

2y³ + 3y² - 6y - 9 = (x + 3/2)(2y² + 6)

 

                         =(2x + 3)(y² + 3)