Find constants A and B such that (3x - 17)/((x - 3)(x - 7) = A/(x - 3) + B/(x - 7).
+23245 (3x - 17)/((x - 3)(x - 7) = A/(x - 3) + B/(x - 7)
Rewrite each fraction with the denominator: (x - 3)(x - 7)
[ A/(x - 3) ] · [ (x - 7)/(x - 7) ] = [ A · (x - 7) ] / [ (x - 3)(x - 7) ] = [ Ax - 7A ] / [ (x - 3)(x - 7) ]
[ B(x - 7) ] · [ (x - 3)/(x - 3) ] = [ B · (x - 3) ] / [ (x - 3)(x - 7) ] = [ Bx - 3B ] / [ (x - 3)(x - 7) ]
Adding these two expressions together:
[ Ax - 7A ] / [ (x - 3)(x - 7) ] + [ Bx - 3B ] / [ (x - 3)(x - 7) ] = [ Ax - 7A + Bx - 3B ] / [ (x - 3)(x - 7) ]
Comparing the original numerator ( 3x - 17 ) to this numerator ( Ax - 7A + Bx - 3B )
3x - 17 = Ax - 7A + Bx - 3B
3x - 17 = (Ax + Bx) + (- 7A - 3B)
Setting the x-terms equal to each other: 3x = Ax + Bx ---> 3 = A + B
Setting the constants equal to each other: -17 = -7A - 3B ---> 17 = 7A + 3B
Solving for A: A + B = 3 ---> -A - B = -3 ---> -3A - 3B = -9
7A + 3B = 17 ---> 7A + 3B = 17
Solving: 4A = 8 ---> A = 2
Since A + B = 3 ---> 2 + B = 3 ---> B = 1
