To find the partial fraction decomposition of the given expression:
[tex]\[
\frac{16 z^2 - 3 z - 3}{(5 z + 3)(z^2 - 2 z + 3)},
\][/tex]
we follow these steps:
1. Identify the form of the decomposition:
Since the denominator is factored as [tex]\( (5z + 3)(z^2 - 2z + 3) \)[/tex], the partial fraction decomposition will have the form:
[tex]\[
\frac{A}{5z + 3} + \frac{Bz + C}{z^2 - 2z + 3},
\][/tex]
where [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are constants to be determined.
2. Express the original fraction in terms of these constants:
[tex]\[
\frac{16z^2 - 3z - 3}{(5z + 3)(z^2 - 2z + 3)} = \frac{A}{5z + 3} + \frac{Bz + C}{z^2 - 2z + 3}.
\][/tex]
3. Combine the right-hand side over a common denominator:
[tex]\[
\frac{A(z^2 - 2z + 3) + (Bz + C)(5z + 3)}{(5z + 3)(z^2 - 2z + 3)}.
\][/tex]
4. Simplify the numerator:
[tex]\[
A(z^2 - 2z + 3) + (Bz + C)(5z + 3).
\][/tex]
Expand and combine like terms:
[tex]\[
A(z^2 - 2z + 3) = Az^2 - 2Az + 3A,
\][/tex]
[tex]\[
(Bz + C)(5z + 3) = Bz(5z) + Bz(3) + C(5z) + C(3) = 5Bz^2 + 3Bz + 5Cz + 3C.
\][/tex]
Combining these:
[tex]\[
Az^2 - 2Az + 3A + 5Bz^2 + (3B + 5C)z + 3C.
\][/tex]
Grouping the like terms:
[tex]\[
(A + 5B)z^2 + (-2A + 3B + 5C)z + (3A + 3C).
\][/tex]
5. Equate coefficients to solve for [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
Compare with the numerator [tex]\( 16z^2 - 3z - 3 \)[/tex]:
[tex]\[
A + 5B = 16,
\][/tex]
[tex]\[
-2A + 3B + 5C = -3,
\][/tex]
[tex]\[
3A + 3C = -3.
\][/tex]
6. Solve the system of linear equations:
From [tex]\( 3A + 3C = -3 \)[/tex]:
[tex]\[
A + C = -1 \quad \text{(i)}.
\][/tex]
From [tex]\( A + 5B = 16 \)[/tex]:
[tex]\[
A = 16 - 5B \quad \text{(ii)}.
\][/tex]
Substitute [tex]\( A \)[/tex] from (ii) into (i):
[tex]\[
(16 - 5B) + C = -1,
\][/tex]
[tex]\[
C = -17 + 5B \quad \text{(iii)}.
\][/tex]
Substitute [tex]\( A \)[/tex] and [tex]\( C \)[/tex] into the second equation:
[tex]\[
-2(16 - 5B) + 3B + 5(-17 + 5B) = -3,
\][/tex]
[tex]\[
-32 + 10B + 3B - 85 + 25B = -3,
\][/tex]
[tex]\[
38B - 117 = -3,
\][/tex]
[tex]\[
38B = 114,
\][/tex]
[tex]\[
B = 3.
\][/tex]
Substitute [tex]\( B = 3 \)[/tex] back into (ii) and (iii):
[tex]\[
A = 16 - 5(3) = 1,
\][/tex]
[tex]\[
C = -17 + 5(3) = -2.
\][/tex]
7. Construct the partial fractions:
[tex]\[
\frac{1}{5z + 3} + \frac{3z - 2}{z^2 - 2z + 3}.
\][/tex]
Thus, the correct partial fraction decomposition is:
[tex]\[
\boxed{\frac{1}{5z + 3}} + \boxed{\frac{3z - 2}{z^2 - 2z + 3}}.
\][/tex]
