posted by Jay on .
Can somebody explain to me why equating coefficients work?
8x^3+13x = Ax^3 + 2Ax + Bx^2 + 2B + Cx + D
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)
(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,