Find constants a, b, and c given that: 6x^3+7x^2-19x+7=(2x-1)(ax^2+bx+c) for all x.

Answer 1

You can just do the long division, or note that
(2x-1)(ax^2+bx+c) = 2ax^3 + (-a+2b)x^2 + (-b+2c)x - c
For this to be identical to 6x^3+7x^2-19x+7, you need
2a = 6
-a+2b = 7
-b+2c = -19
-c = 7
Clearly, a=3, c = -7
and I'm sure you can now find b.