Math Help - [SOLVED] Polynomial factor theorem clarity needed

Example
(x-3) is a factor of f(x)=3x^3-7x^2-5x-3.
So (x-3) is a factor of 3x^3-7x^2-5x-3=(x-3)(bx^2+cx+d).
a=3, b=3,c=2,d=1

I want to clarify my understanding of the below:

a=3 from 3x^3???
considering coefficient x^3 gives b=3....from 3x^3???
considering coefficient x^0 gives -3=-3d..:.d=1. from -3)+d)??? ie the last term??.
considering coefficient x^2 gives -7=c-3b.:.c=2....why c-3b???

Originally Posted by Neverquit

Example
(x-3) is a factor of f(x)=3x^3-7x^2-5x-3.
So (x-3) is a factor of 3x^3-7x^2-5x-3=(x-3)(bx^2+cx+d).
a=3, b=3,c=2,d=1

(x-3) is a factor of 3x^3-7x^2-5x-3
The other factor is a polynomial (degree = 3-1 = 2)

Therefore 3x^3-7x^2-5x-3=(x-3)(ax^2+bx+c)
Expanding gives 3x^3-7x^2-5x-3 = ax^3 + (-3a+b)x^2 + (-3b+c)x - 3c

Identifying the same power terms
a=3
-3a+b=-7
-3b+c=-5
-3c=-3

Therefore
a=3
c=1
b=2