Solve the partial fraction 4X+5/(X-1)(X^2+2X+4)
To solve the partial fraction decomposition of the expression (4X + 5) / [(X - 1)(X^2 + 2X + 4)], we need to determine the unknown constants A, B, and C such that:
(4X + 5) / [(X - 1)(X^2 + 2X + 4)] = A / (X - 1) + (B*X + C) / (X^2 + 2X + 4)
To find the values of A, B, and C, we need to equate the numerators:
4X + 5 = A(X^2 + 2X + 4) + (B*X + C)(X - 1)
Expanding the right side of the equation:
4X + 5 = A*X^2 + 2A*X + 4A + B*X^2 - B*X + C*X - C
Combining like terms:
4X + 5 = (A + B)X^2 + (2A - B + C)X + (4A - C)
By equating the coefficients of the like terms on both sides of the equation, we get the following system of equations:
A + B = 0 (coefficients of X^2)
2A - B + C = 4 (coefficients of X)
4A - C = 5 (constant terms)
Solving this system of linear equations, we find:
A = 1
B = -1
C = -6
Therefore, the partial fraction decomposition is:
(4X + 5) / [(X - 1)(X^2 + 2X + 4)] = 1 / (X - 1) - (X + 6) / (X^2 + 2X + 4)