calculus(equating coefficients)

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!

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  1. 8x^3+13x ≡ Ax^3 + 2Ax + Bx^2 + 2B + Cx + D ...(1)
    =>
    (8-A)x^3-Bx²+(13-2A-C)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 left-hand side of (2) are zero, which then implies
    8-A=0, or A=8,
    etc.

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