5x+1/x^2(x^2+4) decompose into partial fractions
(5x+1) / x^2(x^2+4) = A/x + B/x^2 + (Cx+D)/(x^2+4
5x+1 = Ax(x^2+4) + B(x^2+4) + (Cx+D)x^2
that gives you
Ax^3 + 4Ax + Bx^2 + 4B + Cx^3 + Dx^2 = 5x+1
(A+C)x^3 + (B+D)x^2 + 4Ax + 4B = 5x+1
4A = 5
4B = 1
A+C=0
B+D=0
solve those and you wind up with
(5x+1) / x^2(x^2+4) = (5/4)/x + (1/4)/x^2 + ((-5x-1)/4)/(x^2+4)
or
5/(4x) + 1/(4x^2) - (5x+1)/(4(x^2+4))
5/4x+1/4x^2-5x+1/4(x^2+4)
3 and 8
To decompose the rational function 5x + 1 / [x^2(x^2 + 4)] into partial fractions, follow these steps:
Step 1: Factorize the denominator.
The denominator can be factored as x^2(x^2 + 4) = x^2(x + 2i)(x - 2i), where i is the imaginary unit.
Step 2: Express the fraction in terms of partial fractions.
We express the given fraction as a sum of partial fractions as:
5x + 1 / [x^2(x + 2i)(x - 2i)] = A/x + B/x^2 + C/(x + 2i) + D/(x - 2i)
Step 3: Find the values of the unknown constants.
To find the values of A, B, C, and D, we need to clear the fraction on the right-hand side. Multiply both sides of the equation by the common denominator:
5x + 1 = A(x^2)(x - 2i) + B(x^2)(x + 2i) + Cx(x - 2i) + Dx(x + 2i)
Step 4: Solve for the unknown constants.
Now, we can equate the coefficients of like terms on both sides of the equation. Since the right-hand side has higher powers of x, we can focus on the terms involving x^2, x, and constants.
The x^2 terms:
0x^2 = Ax^3 - 2iAx^2 + Bx^3 + 2iBx^2
Simplifying: (A + B)x^3 + (-2iA + 2iB)x^2 = 0x^2
We get A + B = 0 and -2iA + 2iB = 0.
The x terms:
5x = Cx^2 - 2iCx + Dx^2 + 2iDx
Simplifying: Cx^2 + Dx^2 + (-2iC + 2iD)x = 5x
We get C + D = 0 and -2iC + 2iD = 5.
The constant terms:
1 = -2iAD + 2iBC
We get -2iAD + 2iBC = 1.
Now, we can solve the system of equations formed by the above equations simultaneously to find the values of A, B, C, and D.
Step 5: Solve the system of equations and find the unknown constants.
Solving the system of equations, we find A = -1/4, B = 1/4, C = 1/16i, and D = -1/16i.
Step 6: Write the partial fraction decomposition.
Now that we have the values of the unknown constants, we can write the partial fraction decomposition of the given rational function as:
5x + 1 / [x^2(x^2 + 4)] = -1/4x + 1/4x^2 + 1/16i(x + 2i) - 1/16i(x - 2i)
Therefore, the partial fraction decomposition of 5x + 1 / [x^2(x^2 + 4)] is -1/4x + 1/4x^2 + 1/16i(x + 2i) - 1/16i(x - 2i).
Note: The equation involves complex numbers since the denominator has complex roots.