In Example 9, change the 2x − 3 to 2x + 3 and then determine whether 2x + 3 is a factor.

Data from Example 9:

By using synthetic division, determine whether 2x − 3 is a factor of 2x³ − 3x² + 8x −12.

We first note that the coefficient of x in the possible factor is not 1. Thus, we cannot use r = 3, because the factor is not of the form x − r. However 2x − 3 = 2(x − 3/2) which means that if 2(x − 3/2) is a factor. If we use r = 3/2 and find that the remainder is zero then x − 3/2 is a factor 

Because the remainder is zero, x − 3/2 is a factor. Also, the quotient is 2x² + 8, which may be factored into 2(x² + 4). Thus, 2 is also a factor of the function. This means that 2(x − 3/2) is a factor of the function, and this in turn means that 2x − 3 is a factor. This tells us that 2x³ − 3x² + 8x − 12 = (2x − 3) (x² + 4)

2 -3 8 30 08 2 -12 122 12 0