Problem 16 - A Graphical Approach to College Algebra

Question

Find the partial fraction decomposition for each rational expression. $$\frac{6 x^{5}+7 x^{4}-x^{2}+2 x}{3 x^{2}+2 x-1}$$

Step-by-Step Solution

Answer

Perform long division, decompose remainder using factors (3x-1) and (x+1).

Step 1: Check the Degree of the Polynomials

Check the degree of the numerator (which is 5) and the degree of the denominator (which is 2). Since the degree of the numerator is greater than the degree of the denominator, the first step is to perform polynomial long division.

Step 2: Perform Polynomial Long Division

Divide the numerator by the denominator using polynomial long division. This will express the rational expression as a sum of a polynomial and a proper fraction. Perform the division to find:\[ \frac{6x^5 + 7x^4 - x^2 + 2x}{3x^2 + 2x - 1} = 2x^3 + \frac{R(x)}{3x^2 + 2x - 1} \] where \( R(x) = 40x^4 + 13x^3 - 2x + 0 \).

Step 3: Finding Partial Fractions of the Proper Fraction

Factor the denominator of the remaining fraction if possible. In this case, assume \( 3x^2 + 2x - 1 \) can be factored as \((3x-1)(x+1)\). The partial fractions would take the form:\[ \frac{Ax + B}{3x - 1} + \frac{C}{x + 1} \]

Step 4: Solve for Coefficients

Multiplying through by the common denominator and equating the numerators:\[ R(x) = (Ax + B)(x + 1) + C(3x - 1) \] Compare coefficients to find \( A, B, \) and \( C \). Solve the resulting system of equations to find the values of these coefficients.

Step 5: Write the Full Decomposition

Substitute the values of \( A, B, \) and \( C \) back into the expression obtained from the partial fraction decomposition. The complete partial fraction decomposition will then be:\[ 2x^3 + \frac{Ax + B}{3x - 1} + \frac{C}{x + 1} \] where you have specific values for \( A, B, \) and \( C \).

Step 6: Verify Final Expression

Substitute the values obtained for \( A, B, \) and \( C \) into the partial fraction expression, then combine and simplify to ensure it matches the original expression when added to the quotient from Step 2. This step ensures the decomposition is correct.

Key Concepts

Polynomial Long Division

Polynomial long division helps in simplifying complex rational expressions where the degree of the numerator is higher than that of the denominator. It's similar to numerical long division, but with polynomials.
When applying this technique, the goal is to divide the leading term of the numerator by the leading term of the denominator. You perform repeated steps of multiplication and subtraction till the degree of the remainder is less than that of the divisor.
For instance, when dividing \( 6x^5 + 7x^4 - x^2 + 2x \) by \( 3x^2 + 2x - 1 \), you follow these steps:

• First, divide \( 6x^5 \) by \( 3x^2 \), getting \( 2x^3 \).
• Then, multiply the entire divisor by this quotient \( 2x^3 \).
• Subtract the result from the initial polynomial, yielding a new polynomial of lower degree.
• Repeat these steps until all terms are dealt with.

The end result will be a sum of a polynomial and a rational expression. Here, it turns out to be\( 2x^3 + \frac{R(x)}{3x^2 + 2x - 1} \). After that, you can proceed with partial fraction decomposition.

Degree of Polynomials

The degree of a polynomial is the highest power of the variable in the polynomial expression. It's a critical aspect in understanding how to break down or simplify polynomials, especially in complex algebraic expressions.
For example, in the polynomial \( 6x^5 + 7x^4 - x^2 + 2x \), the highest power of \( x \) is 5, so it is a degree 5 polynomial. Similarly, \( 3x^2 + 2x - 1 \) is a degree 2 polynomial.
Understanding the degree is crucial for:

• Deciding whether polynomial long division is necessary before decomposition.
• Analyzing the behavior and solutions of polynomials.
• Choosing appropriate methods for solving or simplifying expressions.

In our exercise, noticing the degree difference indicated the necessity for polynomial division before moving on to any further simplification or transformation procedures.

Rational Expression

A rational expression is a fraction where both the numerator and the denominator are polynomials. These expressions often need to be simplified or decomposed into simpler terms for ease of computation or integration.
In the exercise, the given expression \( \frac{6x^5 + 7x^4 - x^2 + 2x}{3x^2 + 2x - 1} \) represents a rational expression.
Rational expressions are useful in various fields of mathematics due to their properties:

• They can often be broken into simpler components to analyze or solve equations.
• Understanding how to manipulate them is key in calculus, algebra, and engineering.
• They can model a variety of real-world phenomena due to their inherent flexibility.

Breaking down a complex rational expression often involves techniques like polynomial long division and partial fraction decomposition.

Equation Solving

Equation solving involves finding the values of variables that satisfy the given algebraic equation. In the context of partial fraction decomposition, you're solving for the coefficients to express a complex rational expression in simpler terms.
In our exercise, after simplification by polynomial division, you're left with finding exact values for unknown coefficients \( A, B, \) and \( C \). This is achieved by equating the numerators of equivalent rational expressions and solving the resulting system of equations.
This process involves:

• Setting up an equation based on the given rational expression structure, like \( (Ax + B)(x + 1) + C(3x - 1) = R(x) \).
• Simplifying consolidated expressions to find comparable terms.
• Solving simultaneous equations to determine coefficient values accurately.

Once the values of coefficients are found, they can be substituted back to complete the partial fraction decomposition, which aids in solving related calculus or algebra problems.