Question

Use the division algorithm to rewrite each improper rational expression as the sum of a polynomial and a proper rational expression. Find the partial fraction decomposition of the proper rational expression. Finally, express the improper rational expression as the sum of a polynomial and the partial fraction decomposition. $$ \frac{x^{3}-3 x^{2}+1}{x^{2}+5 x+6} $$

Short answer

\(\frac{x^3 - 3x^2 + 1}{x^2 + 5x + 6} = x - 8 + \frac{109}{x + 2} - \frac{69}{x + 3}\)

Step-by-step solution

Perform Polynomial Long Division

Divide the numerator \(x^3 - 3x^2 + 1\) by the denominator \(x^2 + 5x + 6\). Perform polynomial long division to find the quotient and the remainder.

Identify Quotient and Remainder

By performing the division, we find that the quotient is \(x - 8\) and the remainder is \(40x - 47\). Thus, we can rewrite the expression as follows: \[\frac{x^3 - 3x^2 + 1}{x^2 + 5x + 6} = x - 8 + \frac{40x - 47}{x^2 + 5x + 6}\]

Express the Remainder as a Proper Rational Expression

Rewrite the proper rational expression \(\frac{40x - 47}{x^2 + 5x + 6}\) as the partial fraction decomposition. First, factor the denominator: \(x^2 + 5x + 6 = (x + 2)(x + 3)\).

Set Up Partial Fractions

Set up the equation for partial fractions: \[\frac{40x - 47}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3}\]. Determine the values of \(A\) and \(B\).

Solve for A and B

Multiply through by the denominator to clear the fractions: \[40x - 47 = A(x + 3) + B(x + 2)\]. Equate the coefficients to get a system of equations. 1. \(A + B = 40\)2. \(3A + 2B = -47\)Solve this system to get \(A = 109\) and \(B = -69\).

Write the Partial Fraction Decomposition

Using the values of \(A\) and \(B\), substitute back into the partial fractions: \[\frac{40x - 47}{x^2 + 5x + 6} = \frac{109}{x + 2} + \frac{-69}{x + 3}\].

Combine Polynomial and Partial Fractions

Finally, express the original improper rational expression as the sum of the polynomial quotient and the partial fraction decomposition: \[\frac{x^3 - 3x^2 + 1}{x^2 + 5x + 6} = x - 8 + \frac{109}{x + 2} - \frac{69}{x + 3}\].

Key concepts

polynomial long division

Polynomial long division is the process used to divide a polynomial by another polynomial, similar to long division of numbers. This method helps in breaking down an improper rational expression.

To perform polynomial long division, follow these steps:

• Divide the first term of the numerator by the first term of the denominator.
• Multiply the entire denominator by this result and subtract this product from the numerator.
• Repeat the process with the new polynomial until the degree of the remainder is less than the degree of the divisor.

The result of this process gives you a quotient and a remainder. For the given exercise, dividing the numerator \(x^3 - 3x^2 + 1\) by the denominator \(x^2 + 5x + 6\) yields the quotient \(x - 8\) and the remainder \(40x - 47\). So, you can write the expression as:
\[ \frac{x^3 - 3x^2 + 1}{x^2 + 5x + 6} = x - 8 + \frac{40x - 47}{x^2 + 5x + 6} \].

improper rational expression

An improper rational expression is a fraction where the degree of the numerator is greater than or equal to the degree of the denominator. In our example, \( \frac{x^3 - 3x^2 + 1}{x^2 + 5x + 6} \), the numerator has a degree of 3 while the denominator has a degree of 2, making it an improper fraction.

To simplify an improper rational expression, polynomial long division is often employed. This results in a quotient (which is a polynomial) and a remainder that forms a proper rational expression. A proper rational expression has a numerator with a degree less than that of its denominator.

By converting the improper rational expression into the sum of a polynomial and a proper rational expression, it becomes easier to work with, especially in further applications like integration or solving algebraic equations.

partial fractions

Partial fraction decomposition is the process of expressing a proper rational expression as a sum of simpler fractions called partial fractions. This technique is highly useful in calculus and algebra.

Given the proper rational expression \( \frac{40x - 47}{x^2 + 5x + 6} \), we first factor the denominator: \( (x + 2)(x + 3) \). The goal is to write it as:
\[ \frac{40x - 47}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3}. \]

To find the values of \(A\) and \(B\), we clear the fractions by multiplying through by the denominator, resulting in:
\[ 40x - 47 = A(x + 3) + B(x + 2) \].
We then solve the system of equations derived from comparing coefficients:

• \( A + B = 40 \)
• \( 3A + 2B = -47 \)

This system yields \(A = 109\) and \(B = -69\). Substituting these values back, we get:
\[ \frac{40x - 47}{x^2 + 5x + 6} = \frac{109}{x + 2} - \frac{69}{x + 3} \].
Finally, combining this with the quotient from the polynomial long division, we rewrite the original improper rational expression as:
\[ \frac{x^3 - 3x^2 + 1}{x^2 + 5x + 6} = x - 8 + \frac{109}{x + 2} - \frac{69}{x + 3} \].