Worksheet titled "Partial Fraction Worksheet" containing 15 problems that ask students to express given fractions as the sum of two or more fractions with simpler denominators.

Problem: Partial Fraction Decomposition

The task is to express each of the given fractions as the sum of two or more fractions with simpler denominators. This process is known as partial fraction decomposition. Below, I will solve a few representative problems step by step to illustrate the method.

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General Steps for Partial Fraction Decomposition:

1. Factor the denominator: Factor the denominator into linear and irreducible quadratic factors.
2. Set up the partial fraction form:
- For each distinct linear factor \((x - a)\), write a term of the form \(\frac{A}{x - a}\).
- For repeated linear factors \((x - a)^n\), write terms \(\frac{A_1}{x - a} + \frac{A_2}{(x - a)^2} + \cdots + \frac{A_n}{(x - a)^n}\).
- For irreducible quadratic factors \((ax^2 + bx + c)\), write a term of the form \(\frac{Bx + C}{ax^2 + bx + c}\).
3. Combine the fractions: Multiply through by the common denominator to eliminate the fractions.
4. Solve for the coefficients: Equate the numerators and solve for the constants (e.g., \(A\), \(B\), \(C\)).

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Example Solutions

#### Problem 1:
$$
\frac{x}{x^2 + 5x + 6}
$$

1. Factor the denominator:
$$
x^2 + 5x + 6 = (x + 2)(x + 3)
$$

2. Set up the partial fraction decomposition:
$$
\frac{x}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3}
$$

3. Combine the fractions:
$$
\frac{x}{(x + 2)(x + 3)} = \frac{A(x + 3) + B(x + 2)}{(x + 2)(x + 3)}
$$
Equating the numerators:
$$
x = A(x + 3) + B(x + 2)
$$

4. Expand and collect like terms:
$$
x = Ax + 3A + Bx + 2B
$$
$$
x = (A + B)x + (3A + 2B)
$$

5. Equate coefficients:
- Coefficient of \(x\): \(A + B = 1\)
- Constant term: \(3A + 2B = 0\)

6. Solve the system of equations:
$$
\begin{cases}
A + B = 1 \\
3A + 2B = 0
\end{cases}
$$
From the first equation: \(B = 1 - A\).
Substitute into the second equation:
$$
3A + 2(1 - A) = 0
$$
$$
3A + 2 - 2A = 0
$$
$$
A = -2
$$
Then:
$$
B = 1 - (-2) = 3
$$

7. Write the final decomposition:
$$
\frac{x}{x^2 + 5x + 6} = \frac{-2}{x + 2} + \frac{3}{x + 3}
$$

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#### Problem 4:
$$
\frac{1}{x^2 - 1}
$$

1. Factor the denominator:
$$
x^2 - 1 = (x - 1)(x + 1)
$$

2. Set up the partial fraction decomposition:
$$
\frac{1}{(x - 1)(x + 1)} = \frac{A}{x - 1} + \frac{B}{x + 1}
$$

3. Combine the fractions:
$$
\frac{1}{(x - 1)(x + 1)} = \frac{A(x + 1) + B(x - 1)}{(x - 1)(x + 1)}
$$
Equating the numerators:
$$
1 = A(x + 1) + B(x - 1)
$$

4. Expand and collect like terms:
$$
1 = Ax + A + Bx - B
$$
$$
1 = (A + B)x + (A - B)
$$

5. Equate coefficients:
- Coefficient of \(x\): \(A + B = 0\)
- Constant term: \(A - B = 1\)

6. Solve the system of equations:
$$
\begin{cases}
A + B = 0 \\
A - B = 1
\end{cases}
$$
Add the equations:
$$
2A = 1 \implies A = \frac{1}{2}
$$
Subtract the equations:
$$
2B = -1 \implies B = -\frac{1}{2}
$$

7. Write the final decomposition:
$$
\frac{1}{x^2 - 1} = \frac{\frac{1}{2}}{x - 1} + \frac{-\frac{1}{2}}{x + 1}
$$
Simplify:
$$
\frac{1}{x^2 - 1} = \frac{1}{2(x - 1)} - \frac{1}{2(x + 1)}
$$

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#### Problem 9:
$$
\frac{6}{x^4 - 5x^2 + 4}
$$

1. Factor the denominator:
$$
x^4 - 5x^2 + 4 = (x^2 - 4)(x^2 - 1) = (x - 2)(x + 2)(x - 1)(x + 1)
$$

2. Set up the partial fraction decomposition:
$$
\frac{6}{(x - 2)(x + 2)(x - 1)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{C}{x - 1} + \frac{D}{x + 1}
$$

3. Combine the fractions:
$$
\frac{6}{(x - 2)(x + 2)(x - 1)(x + 1)} = \frac{A(x + 2)(x - 1)(x + 1) + B(x - 2)(x - 1)(x + 1) + C(x - 2)(x + 2)(x + 1) + D(x - 2)(x + 2)(x - 1)}{(x - 2)(x + 2)(x - 1)(x + 1)}
$$
Equating the numerators:
$$
6 = A(x + 2)(x - 1)(x + 1) + B(x - 2)(x - 1)(x + 1) + C(x - 2)(x + 2)(x + 1) + D(x - 2)(x + 2)(x - 1)
$$

4. Solve for \(A\), \(B\), \(C\), and \(D\) using substitution or equating coefficients (details omitted for brevity).

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Final Answers for Selected Problems:

1. $$
\boxed{\frac{x}{x^2 + 5x + 6} = \frac{-2}{x + 2} + \frac{3}{x + 3}}
$$
4. $$
\boxed{\frac{1}{x^2 - 1} = \frac{1}{2(x - 1)} - \frac{1}{2(x + 1)}}
$$

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If you need solutions for other specific problems, please let me know!