Find constants $A$ and $B$ such that \[\frac{x - 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}\]for all $x$ such that $x\neq -1$ and $x\neq 2$. Give your answer as the ordered pair $(A,B)$.

Guest Aug 18, 2022

#1**+1 **

Note that x^2 - x - 2 can be factored as ( x -2) (x + 1)

Multiply through by this factorization and we have

x -7 = A(x + 1) + B( x -2) simplify

x - 7 = (A + B)x + ( A - 2B) equating terms we have this system

A + B = 1

A -2B = -7

subtract the second equation from the first and we have

3B = 8

B = 8/3

and

A + B = 1

A + 8/3 = 1

A = 1 - 8/3

A = - 5/3

CPhill Aug 18, 2022