Problem 41 - Algebra and Trigonometry

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

Divide, decompose, and solve for constants to find the final expression.

Step 1: Check the Degree of Numerator and Denominator

The degree of the numerator \(x^{5}-2x^{4}+x^{3}+x+5\) is 5, and the degree of the denominator \(x^{3}-2x^{2}+x-2\) is 3. Since the degree of the numerator is higher than the degree of the denominator, we must perform polynomial long division.

Step 2: Perform Polynomial Long Division

Divide \(x^5 - 2x^4 + x^3 + x + 5\) by \(x^3 - 2x^2 + x - 2\). Start by dividing the leading term of the numerator \(x^5\) by the leading term of the denominator \(x^3\), which gives you \(x^2\). Multiply the entire divisor by \(x^2\) and subtract from the current dividend, repeat until the degree of the remainder is less than the degree of the divisor.

Step 3: Express Remainder as a Fraction

The result of the long division gives \(x^2 + 0x + 0\) with a remainder. The expression becomes: \(x^2 + \frac{remainder}{x^3-2x^2+x-2}\). Proceed with finding the partial fraction decomposition for the remainder.

Step 4: Factor the Denominator

Factor the denominator \(x^3 - 2x^2 + x - 2\). Use synthetic division or factoring techniques, such as the Rational Root Theorem and polynomial division, to determine the factors as \((x-1)(x^2-x+2)\).

Step 5: Set Up Partial Fraction Decomposition

Since the denominator factors to \((x-1)(x^2-x+2)\), set up the partial fraction decomposition as: \[\frac{A}{x-1} + \frac{Bx+C}{x^2-x+2}\].

Step 6: Find Constants for Decomposition

Multiply through by the factored denominator to eliminate fractions and equate coefficients to solve for \(A\), \(B\), and \(C\). Substitute different values of \(x\) to simplify the calculation. You will find that \(A = \), \(B = \), and \(C = \).

Step 7: Combine Results

Combine the results of the long division and the partial fraction decomposition to express the original rational function as:\[x^2 + \frac{A}{x-1} + \frac{Bx+C}{x^2-x+2}\].

Key Concepts

Polynomial Long Division

Polynomial long division is like the long division we perform with numbers, but instead, we use polynomials. This method helps us divide a polynomial (the dividend) by another polynomial (the divisor) and find a quotient and a remainder.
Steps to Perform Polynomial Long Division

• Identify the leading term of both the dividend and the divisor. The leading term of a polynomial is the term with the highest degree.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this first term of the quotient. Then subtract this product from the current dividend.
• Repeat the division process with the new polynomial (remainder) until the degree of the new dividend is less than the degree of the divisor.

In our case of dividing \(x^5 - 2x^4 + x^3 + x + 5\) by \(x^3 - 2x^2 + x - 2\), we start with dividing \(x^5\) by \(x^3\), which gives us \(x^2\) as the first term of the quotient. Continue the process until the remainder is smaller than the divisor.

Rational Function

A rational function is a type of function that is the ratio of two polynomials. This means the function takes the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not equal to zero.
Key Points to Understand Rational Functions

• The degree of a rational function is determined by the highest power of \(x\) in the numerator and the denominator.
• If the degree of the numerator is higher than that of the denominator, polynomial long division can simplify the function.
• Rational functions can have asymptotes, which are lines that the graph of the function approaches but never touches.

In our exercise, the rational function \(\frac{x^5 - 2x^4 + x^3 + x + 5}{x^3 - 2x^2 + x - 2}\) had a numerator of higher degree than the denominator, necessitating polynomial long division.

Factorization

Factorization, in terms of polynomials, is the process of breaking down a polynomial into a product of simpler polynomials. This can often make the process of analyzing or solving the polynomial easier.
Steps for Factorization

• Look for common factors in all terms of the polynomial and factor them out.
• Recognize patterns, such as the difference of squares, perfect square trinomials, or other recognizable forms.
• Use more advanced methods like synthetic division or the rational root theorem if simpler methods don't work.

In the given exercise, we factor the denominator \(x^3 - 2x^2 + x - 2\) into \((x-1)(x^2-x+2)\) which is essential for setting up the partial fraction decomposition.

Rational Root Theorem

The Rational Root Theorem is a powerful tool for finding the possible rational roots of a polynomial equation. It states that any rational solution \(\frac{p}{q}\) of a polynomial equation is a factor of the constant term divided by a factor of the leading coefficient.
How to Use the Rational Root Theorem

• Write down all factors of the constant term (the term without \(x\)).
• Write down all factors of the leading coefficient (the coefficient of the term with the highest degree).
• Form all possible fractions \(\frac{p}{q}\) using these factors, simplifying them when possible.
• Test these possible roots in the polynomial to determine which are actual roots.

By applying the rational root theorem to the polynomial \(x^3 - 2x^2 + x - 2\), we effectively found that one of its factors is \(x - 1\). This reveals one possible root, assisting in the factorization process.