Solve the partial fraction 3X^2+3X-2/(X^2-1)(X+3)=P+Q/(X-1)+R/(X+1) and then evaluate P+Q+R
To solve the partial fraction, let's start by factoring the denominator:
(X^2-1)(X+3) = (X-1)(X+1)(X+3).
Now, we can rewrite the expression:
(3X^2 + 3X - 2) / ((X-1)(X+1)(X+3)) = P / (X-1) + Q / (X+1) + R / (X+3).
To find the values of P, Q, and R, we'll combine the fractions on the right side:
(3X^2 + 3X - 2) = P(X+1)(X+3) + Q(X-1)(X+3) + R(X-1)(X+1).
Expanding the right side of the equation:
3X^2 + 3X - 2 = (P(X^2 + 4X + 3) + Q(X^2 + 2X - 3) + R(X^2 - 1).
Combine like terms:
3X^2 + 3X - 2 = (P + Q + R)X^2 + (4P + 2Q)X + (3P - 3Q - R).
Now, we can equate the coefficients of X^2, X, and the constant term on both sides of the equation:
Coefficient of X^2: P + Q + R = 3.
Coefficient of X: 4P + 2Q = 3.
Constant term: 3P - 3Q - R = -2.
We now have a system of three equations with three unknowns:
P + Q + R = 3,
4P + 2Q = 3,
3P - 3Q - R = -2.
Solving this system of equations, we find:
P = 1, Q = -1, R = 3.
Therefore, P + Q + R = 1 + (-1) + 3 = 3.