Problem 45 - College Algebra Essentials

Question

perform each long division and write the partial fraction decomposition of the remainder term. $$ \frac{x^{4}-x^{2}+2}{x^{3}-x^{2}} $$

Step-by-Step Solution

Answer

The solution to the given polynomial long division and partial fraction decomposition is \( x + \frac{2}{x^3 - x^2} \). The partial fraction decomposition of the remainder is \( \frac{2}{x^2(x-1)} \), which doesn't break down any further because the factors of the denominator are irreducible.

Step 1: Polynomial Long Division

Here we will perform the long division of the given polynomials: \(x^{4}-x^{2}+2\) by \(x^{3}-x^{2}\). First, divide the leading term in the numerator by the leading term in the denominator, which gives \(x\). Multiply \(x\) by the denominator to subtract it from the numerator. Perform the subtraction and bring down the next term from the numerator. Repeat this process until all terms have been brought down. The result is our quotient plus the remainder divided by the original denominator.

Step 2: Partial Fraction Decomposition

The result from the first step should provide a term that is the sum of two fractions (the quotient and the remainder over the original denominator). In the second term, shift the denominator to the left to make it equal to 1. The polynomial now looks like a sum of fractions with simpler denominators. Break apart the remaining fraction with multiple terms in the denominator. This results in the partial fraction decomposition.

Step 3: Final Solution

Write the final solution. This should include the original fraction represented as a sum of simpler fractions that are equivalent to the original.

Key Concepts

Polynomial Long Division

Polynomial long division is similar to the long division process with numbers, but it's applied to polynomials. When finding the partial fraction decomposition, it's an essential step if the degree of the numerator is greater than or equal to the degree of the denominator.

To perform polynomial long division:

• Identify the leading terms of both the dividend (numerator) and the divisor (denominator).
• Divide the leading term of the dividend by the leading term of the divisor. This becomes the first term of your quotient.
• Multiply the entire divisor by this new term of the quotient. Then, subtract this result from the dividend to get a new polynomial.
• Repeat the process with the new polynomial by bringing down the next term of the original dividend until no terms from the dividend remain.

This will give you a quotient and possibly a remainder. The remainder will need further processing for decomposition into partial fractions.

Remainder Theorem

The Remainder Theorem provides valuable insights when performing division with polynomials. In essence, it states that when a polynomial \( f(x) \) is divided by \( (x-a) \), the remainder of this division is \( f(a) \).

For partial fraction decomposition, understanding the Remainder Theorem helps in identifying and working with the remainder polynomial after long division. If a polynomial \( P(x) \) is divided by \( (x - r) \), and you find yourself left with a remainder that needs further simplification, the Remainder Theorem gives a clear starting point for determining the value at specific points such as \( x = a \).

Using this knowledge, we can easily evaluate where our polynomial gives certain values, which are crucial when we aim to break down complex fractions into simpler, more manageable pieces.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. An understanding of rational functions is key when performing partial fraction decomposition, as the goal is often to simplify them into a sum of simpler fractions.

A rational function will typically be in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. Partial fraction decomposition transforms this into a sum of fractions such as \( \frac{A}{(x - r)} \) or \( \frac{Bx + C}{(x^2 + px + q)} \) when \( Q(x) \) can be factored further.

• Simplifies integration of complex fractions, particularly when dealing with calculus problems.
• Makes the overall algebraic manipulation easier and more intuitive.

Partial fraction decomposition's utility in breaking down rational functions into simpler fractions cannot be overstated, especially when preparing to integrate or solve higher-level mathematical problems.