+145 Find constants A and B such that
{x + 7}/{x^2 - x - 2} = {A}/{x - 2} + {B}/{x + 1}
for all x such that x is not equal to -1 and x is not equal to 2. Give your answer as the ordered pair (A,B).
+129852 (x + 7 ) / ( x^2 - x - 2) = A / (x - 2) + B / ( x + 1)
We can use partial fractions, here
We can write
( x + 7) /[( ( x + 1)( x - 2)] = A / ( x - 2) + B / ( x + 1)
Multiply through by (x + 1) ( x -2)
x + 7 = A ( x + 1) + B( x - 2) simplify
x + 7 = Ax + A + Bx - 2B
x + 7 = ( A + B) x + ( A - 2B)
Equating coefficients, we have
A + B = 1
A - 2B = 7 subtract the first equation from the second
-3B = 6
B = -2
And we can find A as A + B = 1 ⇒ A - 2 = 1 ⇒ A = 3
So (A, B) = ( 3, - 2)
