Find constants A and B such that for all x such that x not equal to 2 and 4, (2x + 3)/(x^2 - 6x + 8) = A/(x - 2) + B/(x - 4). Give your answer as the ordered pair (A,B).
+23245 (2x + 3) / (x2 - 6x + 8) = A / (x - 2) + B / (x- 4)
Since x2 - 6x + 8 can be factored as (x - 2)(x - 4),
multiply each term by (x - 2)(x - 4) to get: 2x + 3 = A(x - 4) + B(x - 2)
2x + 3 = Ax - 4A + Bx - 2B
Set the x-terms equal to each other: 2x = Ax + Bx ---> 2x = (A + B)x ---> A + B = 2
Set the constants equal to each other: 3 = -4A - 2B ---> 4A + 2 B = -3
Solve this set of two equations: A + B = 2 to get your answer.
4A + 2B = -3
