Question

Find the partial fraction decomposition of each rational expression. $$ \frac{x^{2}-11 x-18}{x\left(x^{2}+3 x+3\right)} $$

Short answer

\( \frac{-6}{x} + \frac{7x + 7}{x^2 + 3x + 3} \)

Step-by-step solution

- Factor the Denominator

Factor the denominator of the rational expression. Given \ \ \[ x(x^2 + 3x + 3) \], the denominator is already factored as \( x \) and \( x^2 + 3x + 3 \).

- Set up the Partial Fractions

Set up the form of the partial fraction decomposition based on the factors of the denominator. Since we have linear term \( x \) and an irreducible quadratic \( x^2 + 3x + 3 \), we get: \ \ \[ \frac{x^2 - 11x - 18}{x(x^2 + 3x + 3)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 3x + 3} \]

- Combine the Partial Fractions

Combine the partial fractions on the right-hand side to form a single fraction: \[ \frac{A}{x} + \frac{Bx + C}{x^2 + 3x + 3} = \frac{A(x^2 + 3x + 3) + (Bx + C)x}{x(x^2 + 3x + 3)} \]

- Simplify and Match Numerators

Simplify the numerator of the combined fraction and equate it to the numerator of the original fraction: \ \ \[ A(x^2 + 3x + 3) + (Bx + C)x = A x^2 + 3 A x + 3 A + B x^2 + C x \] \ Since \(A x^2 + 3A x + 3A + B x^2 + C x \) should equal \(x^2 - 11x - 18 \), we get: \ \ \( A + B = 1 \) \ \( 3A + C = -11 \) \ \( 3A = -18 \)

- Solve for A, B, and C

Solve the system of equations from Step 4: \ Solve \( 3A = -18 \) to get \( A = -6 \). \ Substitute \( A = -6 \) into \( A + B = 1 \) to get \( -6 + B = 1 \), thus \( B = 7 \). \ Substitute \( A = -6 \) into \( 3A + C = -11 \) to get \( 3(-6) + C = -11 \), thus \( -18 + C = -11 \) and \( C = 7 \).

- Write the Final Partial Fraction Decomposition

Substitute the found values of A, B, and C into the partial fractions: \ \ \[ \frac{x^2 - 11x - 18}{x(x^2 + 3x + 3)} = \frac{-6}{x} + \frac{7x + 7}{x^2 + 3x + 3} \]

Key concepts

rational expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. These expressions can sometimes be simplified or broken down for easier manipulation, such as through partial fraction decomposition. It's important to make sure the denominator is not zero, as division by zero is undefined.
In our problem, the rational expression given is \( \frac{x^2 - 11x - 18}{x(x^2 + 3x + 3)} \). Here, the numerator is \( x^2 - 11x - 18 \) and the denominator is \( x(x^2 + 3x + 3) \).
Always start by examining both parts of the fraction before proceeding to operations like factoring or decomposition.

factoring the denominator

Factoring is a key step in simplifying rational expressions and finding partial fraction decompositions. Our first task is to factor the denominator completely.
In the expression \( \frac{x^2 - 11x - 18}{x(x^2 + 3x + 3)} \), the denominator is already partially factored as \(x (x^2 + 3x + 3)\).
Factoring plays a crucial role because it tells us what kinds of terms will appear in the partial fraction decomposition. For example, linear factors like \( x \) result in terms like \( \frac{A}{x} \), and quadratic factors like \( x^2 + 3x + 3 \) (which cannot be factored further) result in terms like \( \frac{Bx + C}{x^2 + 3x + 3} \).
Properly factoring the denominator sets the stage for the rest of the decomposition process.

system of equations

When decomposing a rational expression into partial fractions, creating a system of equations is essential for solving for the unknowns in our decomposition.
After setting up the equation \( \frac{x^2 - 11x - 18}{x(x^2 + 3x + 3)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 3x + 3} \), we combine and simplify the right-hand side to a single fraction: \( \frac{A(x^2 + 3x + 3) + (Bx + C)x}{x(x^2 + 3x + 3)} \). By expanding and matching terms between the expanded fraction and the original numerator, we derive a system of equations: - \( A + B = 1 \) - \( 3A + C = -11 \) - \( 3A = -18 \). Each equation correlates to coefficients of identical powers of x in the numerator. This system of equations is crucial for finding the values of A, B, and C.

solving for constants

The final step in partial fraction decomposition is solving for the constants that appear in the fraction terms. Once we have our system of equations, solving it involves basic algebraic techniques.
In our example, the system of equations is: - \( A + B = 1 \) - \( 3A + C = -11 \) - \( 3A = -18 \). First, solve the simplest equation: \( 3A = -18 \). This gives us \( A = -6 \).
Substitute \( A \) into the first equation: \( -6 + B = 1 \), leading to \( B = 7 \).
Finally, substitute \( A \) into the second equation: \( 3(-6) + C = -11 \), resulting in \( -18 + C = -11 \) and thus \( C = 7 \).
Plugging these values back into the partial fractions provides the final decomposition: \( \frac{x^2 - 11x - 18}{x(x^2 + 3x + 3)} = \frac{-6}{x} + \frac{7x + 7}{x^2 + 3x + 3} \).