Representation of a given polynomial in factorial notation

Hi,

Represent f(x)=$ 2x^4 -12x^3 +24x^2 - 30x +9 $ and its successive differences in factorial notation

Solution:Let f(x) =$ 2x^4 - 12x^3 +24x^2 -30x +9 $

=$2x^{(2)} +bx^{(3)} +cx^{(2)} +dx +9 $

=2x(x-1)(x-2)(x-3) +bx(x-1)(x-2) +cx(x-1) +dx +9 where b,c and d are constants to be determined.

Putting x=1, we get,

-7=0+0+0+d+9

$ \therefore $ d=-16

Similarly, putting x=2 and x=3 in turn, we find c=2, b=0

f(x)=$ 2x^{(4)}+2x^{(2)} - 16x^{(1)} + 9 $

$ \Delta f(x)=8x^{(3)} + 4x -16 $

$ \Delta^2 f(x)=24x^{(2)}+ 4 $

$ \Delta^3 f(x)=48x $

$\Delta^4 f(x)= 48 $

Is this correct ?

Reply is appreciated.