Find constants $A$ and $B$ such that
x+7x2−x−2=Ax−2+Bx+1
for all $x$ such that $x\neq -1$ and $x\neq 2$. Give your answer as the ordered pair.
Or if you prefer
https://gyazo.com/cdbf05411074209828b51f33c00774d2
+130474 x + 7 A B
________ = ______ + ____
x^2 - x - 2 x - 2 x + 1
This is a partial fraction decomposition
The denominator on the left can be factored as (x - 2) (x + 1)
Multiply through by this denominator and we get
1x + 7 = A(x + 1) + B(x - 2)
1x + 7 = (A + B) x + A - 2B
Equate coefiicients and we get the system
A + B = 1
A - 2B = 7 subtract the second from the first
3B = - 6
B = - 2
So A = A + -2 = 1 ⇒ A = 3
