Question

Mixed Practice In Problems \(51-58\), 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.. $$ \frac{x^{3}+x^{2}-3}{x^{2}+3 x-4} $$

Short answer

\(1 + \frac{x + 9}{(x - 1)(x + 4)} = 1 + \frac{2}{x - 1} + \frac{7}{x + 4}\)

Step-by-step solution

- Perform Polynomial Long Division

Divide the numerator by the denominator using polynomial long division. Divide \(x^3 + x^2 - 3\) by \(x^2 + 3x - 4\). The quotient will be the polynomial part of the expression.

- Find the Quotient

Divide the first term of the numerator \(x^3\) by the first term of the denominator \(x^2\), which gives \(x\). Multiply \(x\) by the entire denominator and subtract the result from the original numerator to find the new numerator.

- Repeat Division Steps

Repeat the process of dividing the first term of the new numerator by the first term of the denominator. Continue this process until the degree of the remainder is less than the degree of the denominator.

- Write the Division Result

After performing the division, you are left with a quotient (polynomial) and a remainder. Write the result of the division as: \(\text{Quotient} + \frac{\text{Remainder}}{\text{Denominator}}\).

- Find the Partial Fraction Decomposition

Decompose the proper rational expression \(\frac{\text{Remainder}}{x^2 + 3x - 4}\) into partial fractions. Factor the denominator, then express the rational expression as a sum of fractions with these factors as denominators.

- Combine Everything

Combine the polynomial part and the partial fractions to express the original improper rational expression as the sum of a polynomial and the partial fractions.

Key concepts

Polynomial Division

Polynomial division is similar to long division with numbers. It helps us divide a polynomial numerator by a polynomial denominator.
When dealing with polynomials, we focus on dividing the highest degree terms first. Here's how it works:

• Identify the leading terms of the numerator and denominator.
• Divide the leading term of the numerator by the leading term of the denominator.
• Multiply the entire denominator by this result, and subtract it from the numerator.
• Repeat the process with the new numerator until the degree of the numerator is less than the degree of the denominator.

After performing polynomial long division for the expression \(\frac{x^{3}+x^{2}-3}{x^{2}+3 x-4}\), we will get a quotient and a remainder, which we can then decompose further.

Improper Rational Expression

An improper rational expression is where the degree of the numerator is greater than or equal to the degree of the denominator.
In the case of our exercise, \(\frac{x^{3}+x^{2}-3}{x^{2}+3x-4}\) is an improper rational expression because the degree of the numerator (3) is greater than the degree of the denominator (2).

To handle improper rational expressions effectively, we need to:

• First, perform polynomial division to simplify it into a proper form.
• Extract the polynomial part from the quotient obtained.
• Then, decompose the remaining proper fraction using partial fractions.

This process turns a potentially complex problem into smaller, more manageable parts.

Partial Fractions

Partial fraction decomposition simplifies a complicated rational expression into a sum of simpler fractions.
Here's how we do it:

• First, factor the denominator completely.
• Express the proper rational expression as a sum of fractions with these factors as their denominators.
• Solve for the constants in the numerators of these fractions.

In our exercise, after polynomial division, we decompose \(\frac{\text{Remainder}}{x^2 + 3x - 4}\).
Factoring the denominator gives \(x^2 + 3x - 4 = (x - 1)(x + 4)\).
\br>So, we rewrite it as: \(\frac{\text{A}}{(x - 1)} + \frac{\text{B}}{(x + 4)}\).
\br>Solve for A and B to complete the partial fraction decomposition.
\br>Finally, sum the polynomial part and these partial fractions to express the original improper rational expression more simply.