Sure, let's resolve the given expression into partial fractions step by step.
Given the expression:
[tex]\[
\frac{x^3 - 10x^2 + 26x + 3}{(x + 3)(x - 3)^3}
\][/tex]
Step 1: Denominator Factorization
The denominator [tex]\((x + 3)(x - 3)^3\)[/tex] is already factored, which helps in setting up the partial fraction decomposition.
Step 2: Set up the Partial Fraction Decomposition Form
For the given fraction, the form of the partial fractions will be:
[tex]\[
\frac{x^3 - 10x^2 + 26x + 3}{(x + 3)(x - 3)^3} = \frac{A}{x + 3} + \frac{B}{x - 3} + \frac{C}{(x - 3)^2} + \frac{D}{(x - 3)^3}
\][/tex]
where [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex] are constants to be determined.
Step 3: Express the equation in terms of a single fraction
[tex]\[
\frac{A}{x + 3} + \frac{B}{x - 3} + \frac{C}{(x - 3)^2} + \frac{D}{(x - 3)^3} = \frac{A(x-3)^3 + B(x+3)(x-3)^2 + C(x+3)(x-3) + D(x+3)}{(x + 3)(x - 3)^3}
\][/tex]
Step 4: Combine the Numerators
Since the denominators are the same, we can combine the numerators:
[tex]\[
x^3 - 10x^2 + 26x + 3 = A(x - 3)^3 + B(x + 3)(x - 3)^2 + C(x + 3)(x - 3) + D(x + 3)
\][/tex]
Step 5: Expand the Right Side
[tex]\[
A(x - 3)^3 = A(x^3 - 9x^2 + 27x - 27)
\][/tex]
[tex]\[
B(x + 3)(x - 3)^2 = B(x + 3)(x^2 - 6x + 9) = B(x^3 - 6x^2 + 9x + 3x^2 - 18x + 27) = B(x^3 - 3x^2 - 9x + 27)
\][/tex]
[tex]\[
C(x + 3)(x - 3) = C(x^2 - 9)
\][/tex]
[tex]\[
D(x + 3) = Dx + 3D
\][/tex]
Step 6: Combine all terms:
[tex]\[
A(x^3 - 9x^2 + 27x - 27) + B(x^3 - 3x^2 - 9x + 27) + C(x^2 - 9) + Dx + 3D
\][/tex]
Collecting like terms:
[tex]\[
(A + B)x^3 + (-9A - 3B + C)x^2 + (27A - 9B - 9Cx) + (-27A - 27B - 9C + 3D)
\][/tex]
Step 7: Equate coefficients of corresponding powers of [tex]\(x\)[/tex] from both sides of the equation:
[tex]\[
x^3: A + B = 1
\][/tex]
[tex]\[
x^2: -9A - 3B + C = -10
\][/tex]
[tex]\[
x: 27A - 9B + D = 26
\][/tex]
[tex]\[
Constant: -27A + 27B - 9C + 3D = 3
\][/tex]
Step 8: Solve the system of equations
After solving, we get:
[tex]\[
A = \frac{8}{9},\quad B = \frac{1}{9},\quad C = -\frac{5}{3},\quad D = 3
\][/tex]
Step 9: Write out the partial fractions
[tex]\[
\frac{8}{9(x + 3)} + \frac{1}{9(x - 3)} - \frac{5}{3(x - 3)^2} + \frac{3}{(x - 3)^3}
\][/tex]
Thus, the partial fraction decomposition of [tex]\(\frac{x^3 - 10x^2 + 26x + 3}{(x + 3)(x - 3)^3}\)[/tex] is:
[tex]\[
\frac{8}{9(x + 3)} + \frac{1}{9(x - 3)} - \frac{5}{3(x - 3)^2} + \frac{3}{(x - 3)^3}
\][/tex]
