calculus(equating coefficients)
posted by Jay .
Can somebody explain to me why equating coefficients work?
Example:
8x^3+13x = Ax^3 + 2Ax + Bx^2 + 2B + Cx + D
expanded into:
8x^3 + 13x = Ax^3 + Bx^2 + (2A+C)x + 2(B+D)
where A,B,C,D are constants.
Why does 8 = A; 0 = B; 13 = 2A + C; etc.
I know they have same power variables, but why does this actually work? Thanks!

8x^3+13x ≡ Ax^3 + 2Ax + Bx^2 + 2B + Cx + D ...(1)
=>
(8A)x^3Bx²+(132AC)x (2B+C)≡ 0 ...(2)
This is an identity, and has to work for all values of x.
This can happen if and only if the coefficients on the lefthand side of (2) are zero, which then implies
8A=0, or A=8,
etc.