x+P(x−1)(x−3)=Qx−1+2x−3\frac{x+P}{(x-1)(x-3)} = \frac{Q}{x-1} + \frac{2}{x-3}(x−1)(x−3)x+P=x−1Q+x−32
x+P(x−1)(x−3)=Q(x−3)+2(x−1)(x−1)(x−3)\frac{x+P}{(x-1)(x-3)} = \frac{Q(x-3)+2(x-1)}{(x-1)(x-3)}(x−1)(x−3)x+P=(x−1)(x−3)Q(x−3)+2(x−1)
Comparing LHS and RHS of the equation, we have
x+P=Qx−3Q+2x−2x + P = Qx - 3Q + 2x - 2x+P=Qx−3Q+2x−2
P=−3Q−2P = -3Q - 2P=−3Q−2
Q+2=1 ⟹ Q=1−2=−1Q + 2 = 1 \implies Q = 1 - 2 = -1Q+2=1⟹Q=1−2=−1
P=−3(−1)−2=3−2=1P = -3(-1) - 2 = 3 - 2 = 1P=−3(−1)−2=3−2=1
P+Q=1+(−1)=0P + Q = 1 + (-1) = 0P+Q=1+(−1)=0