Problem 43 - Precalculus Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{x^{5}-2 x^{4}+x^{3}+x+5}{x^{3}-2 x^{2}+x-2}$$

Step-by-Step Solution

Answer

Use polynomial long division, then express as a sum of the quotient and partial fractions of the remainder.

Step 1: Check If Proper Rational Function

A rational function is proper if the degree of the numerator is less than the degree of the denominator. Here, the numerator has degree 5 and the denominator has degree 3. Since the degree of the numerator is greater, it is an improper fraction.

Step 2: Perform Polynomial Long Division

Divide \(x^5 - 2x^4 + x^3 + x + 5\) by \(x^3 - 2x^2 + x - 2\). The first term is \(x^2\). Multiply the entire divisor by \(x^2\) and subtract from the original polynomial. Repeat the process with the resulting polynomial. After division, you should find the quotient and remainder.

Step 3: Express the Original Function

After division, let the quotient be \(Q(x)\) and the remainder be \(R(x)\). Then, express the original rational function as: \[ Q(x) + \frac{R(x)}{x^3 - 2x^2 + x - 2} \]

Step 4: Factor the Denominator

The denominator \(x^3 - 2x^2 + x - 2\) may need to be factored further if possible to identify simple factors or irreducible quadratic factors. Use synthetic division or the Rational Root Theorem to find factors, making it simpler to decompose.

Step 5: Set Up Partial Fractions

Assuming the denominator factors into linear or irreducible quadratic terms, set up the partial fractions accordingly. For instance, if it factors into \((x-1)(x-1)(x+1)\), set up partial fractions like: \[ \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1} \]

Step 6: Solve for Coefficients

To solve for coefficients, multiply both sides by the factored denominator to clear the fractions. Compare coefficients on both sides or plug in suitable values for \(x\) to create a system of equations to solve for \(A\), \(B\), and \(C\).

Step 7: Write the Decomposition

Substitute all found coefficients back into the partial fractions. Combine this with any quotient found in Step 3 for the complete decomposed expression.

Key Concepts

Polynomial Long Division

Polynomial Long Division is a fundamental method used to divide polynomials, much like how long division is used with numbers.
This process comes into play when dealing with dividing a polynomial by another polynomial.
It helps us simplify complex expressions and is especially useful when dealing with improper fractions in rational functions.

Here's how it works in a simple way:

• Set up the division: Write the dividend (the polynomial to be divided) and the divisor (the polynomial doing the dividing) as you would with regular numbers.
• Figure out how many times the leading term of the divisor fits into the leading term of the dividend.
  Use the result as the first term of your quotient. Repeat the process.
• Multiply the entire divisor by this new term, write the result under the dividend, and subtract to find the new dividend.
• Continue this until each term of the dividend has been accounted for.

This process helps break down a polynomial into something more manageable, and in the context of Partial Fraction Decomposition, it enables us to transform improper fractions into proper fractions.

Rational Function

A Rational Function is a fraction where both the numerator and the denominator are polynomials.
It is expressed in the form: \[ f(x) = \frac{P(x)}{Q(x)} \]
where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\).

Some important points about rational functions include:

• The domain of a rational function is all real numbers except where the denominator equals zero.
• A rational function is called 'proper' when the degree of the polynomial in the numerator is less than the degree in the denominator.
• It is 'improper' when the numerator's degree is greater than or equal to the denominator's degree, requiring division to simplify.

In the given exercise, the function is improper, leading us to utilize methods like polynomial long division to simplify and analyze the expression.

Improper Fraction

An Improper Fraction in the context of rational functions is when the degree of the numerator is at least as large as that of the denominator.
For example, in the exercise, the numerator has a degree of 5 while the denominator has a degree of 3, making it an improper fraction.

Here’s why it matters:

• You need to convert an improper fraction into a proper fraction.
  This helps in further analysis, such as partial fraction decomposition.
• The conversion involves using polynomial long division to find a quotient and a remainder.
• After division, the result expresses the improper fraction as a sum of the quotient and a proper fraction.

Simplifying an improper fraction to a proper one enables easier handling and decomposition into partial fractions.

Synthetic Division

Synthetic Division is a simplified form of polynomial division that can be easier and faster than polynomial long division.
It is mainly used when dividing by a linear factor of the form \((x - c)\).
This method is efficient for finding the roots of polynomials and is often used when attempting to factor a polynomial.

Here's how it works in simple terms:

• Identify the root \(c\) from the linear factor \((x - c)\).
• Set up a table that includes the coefficients of the polynomial you wish to divide.
• Drop the leading coefficient down, multiply by \(c\), and add to the next coefficient. Repeat this across the row.
• The last value in your computed row represents the remainder, while the other values form the coefficients of the quotient polynomial.

Synthetic division is particularly useful in the Step 4 of partial fraction decomposition as it helps simplify, speed up, and verify polynomial factorization.