Problem 39 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition. Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{x^{2}+5}{(x+1)\left(x^{2}-2 x+3\right)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the given rational expression can be written as \(\frac{A}{x+1} + \frac{Bx + C}{x^{2}-2x+3}\), where the values of A, B, and C are determined by solving a system of linear equations. This solution should then be checked algebraically to confirm its correctness.

Step 1: Identify the Form

Firstly, identify if the given function is in the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and the degree of \(P(x)\) is less than that of \(Q(x)\). In this case, the given function is \(\frac{x^{2}+5}{(x+1)\left(x^{2}-2 x+3\right)}\). It satisfies the condition, so, no long division is required.

Step 2: Separate the Denominator

Next, separate the denominators into individual fractions. The expression becomes: \(\frac{x^{2}+5}{(x+1)\left(x^{2}-2 x+3\right)} = \frac{A}{x+1} + \frac{Bx + C}{x^{2}-2x+3}\) where A, B, and C are constants that need to be determined.

Step 3: Making the Denominators Same

For simplification, make the denominators the same for all terms: \(x^{2}+5 = A(x^{2}-2x+3) + (Bx + C)(x+1)\). Expand this out and collect like terms.

Step 4: Comparing Coefficients

By comparing the coefficients between the left side and the expanded right side of the equation, you get a system of linear equations in variables A, B, C. Solve this system to determine the values of A, B, and C.

Step 5: Form the Final Decomposed Expression

Once you've determined the values of A, B, and C, substitute those values back into the equation you got in Step 2. This will be the decomposed expression for the given fraction.

Step 6: Check the Result Algebraically

Finally, check if you have decomposed the fraction correctly by recomposing the fraction and algebraically simplifying it. This should result in the original function.

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are akin to ratios found with ordinary numbers but involve variables instead. In particular, rational expressions can look complex due to the presence of polynomial forms both above and below the fraction line.

Understanding the basics of rational expressions is crucial because they often appear in algebra and calculus problems. Simplifying them involves breaking down complex expressions into more manageable pieces.

• The numerator is a polynomial, like \(x^2 + 5\).
• The denominator is often a product of factors, as seen in \((x+1)(x^2 - 2x + 3)\).

Key aspects when working with these expressions include ensuring the expression makes sense by determining any restrictions on the variable values, like avoiding division by zero. Rational expressions are also central to techniques like partial fraction decomposition, where they are separated into simpler fractions.

In essence, mastering rational expressions empowers you to manipulate and simplify complex mathematical relationships.

Polynomial Long Division

Polynomial long division is a technique used when the degree of the numerator is greater than or equal to the degree of the denominator. This scenario doesn't apply to every rational expression, but when applicable, it provides a method for simplifying or rewriting the expression.

For our specific exercise, we identified that polynomial long division wasn't necessary because the degree of the numerator \(x^2 + 5\) was smaller than the degree of the denominator. Here’s a simple walkthrough of the process for other cases:

• Identify dividend and divisor: The numerator is your dividend, and the denominator is your divisor.
• Divide, multiply, subtract, bring down: Similar to long division with numbers but with polynomials.
• Repeat until the degree of the remainder is less than the divisor degree.

Polynomial long division enables you to separate the fraction into a whole term plus a simpler fraction, making further operations more straightforward.

Linear Equations

Linear equations emerge often, especially when finding the coefficients needed in partial fraction decomposition. In the process, you may set up and solve a system of linear equations to get these coefficients.

For example, if we have an expression needing coefficients \(A, B,\) and \(C\) to be determined, the constructing and solving of linear equations is vital.

• Set up the equation by expanding and equating like terms.
• Solve for each variable: Often by substitution or elimination methods.
• Validate by plugging back into the original expression.

In partial fraction decomposition, solving for these constants ensures the accuracy of the decomposed expression. Mastering linear equations is fundamental, as this practice aids in logically deconstructing and solving mathematics problems across various topics.