Answer :
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]
[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]
