Problem 16 - Precalculus

Question

(a) factor the denominator of the given \(\mathrm{ra}\) tional expression; (b) determine the form of the partial fraction decomposition for the given rational expression; and (c) determine the values of the constants in the partial fraction decomposition that you gave in part (b). To help you in spotting errors, use the fact that in part (c), each of the required constants turns out to be an integer. $$\frac{11 x^{3}+35 x-7}{x^{4}+6 x^{2}+9}$$

Step-by-Step Solution

Answer

The partial fractions are \( \frac{11x}{x^2 + 3} + \frac{2x - 7}{(x^2 + 3)^2} \).

Step 1: Factor the Denominator

The given rational expression is \( \frac{11x^3 + 35x - 7}{x^4 + 6x^2 + 9} \). First, let's factor the denominator, \( x^4 + 6x^2 + 9 \). Notice that it can be rewritten as a quadratic in terms of \( x^2 \): \((x^2)^2 + 6(x^2) + 9\), which simplifies to \((x^2)^2 + 6x^2 + 9 = (x^2 + 3)^2 \). Hence, the expression factors completely as \((x^2 + 3)^2\).

Step 2: Form Partial Fraction Decomposition

With the denominator factored as \((x^2 + 3)^2\), the partial fraction decomposition takes the form:\[ \frac{11x^3 + 35x - 7}{(x^2 + 3)^2} = \frac{Ax + B}{x^2 + 3} + \frac{Cx + D}{(x^2 + 3)^2} \] Here, \(Ax + B\) and \(Cx + D\) are numerators to be determined.

Step 3: Clear the Denominators

Multiply both sides by \((x^2 + 3)^2\) to clear the denominators:\[ 11x^3 + 35x - 7 = (Ax + B)(x^2 + 3) + (Cx + D) \] Simplify the right side:\[ = Ax^3 + 3Ax + Bx^2 + 3B + Cx + D \] Combine like terms: \[ = Ax^3 + Bx^2 + (3A + C)x + (3B + D) \]

Step 4: Compare Coefficients

Equate coefficients from both sides:- For \(x^3\): \(A = 11\)- For \(x^2\): \(B = 0\)- For \(x\): \(3A + C = 35\)- For the constant: \(3B + D = -7\)

Step 5: Solve for Constants

From \(A = 11\) and \(B = 0\), substitute into other equations:- \(3(11) + C = 35 \implies 33 + C = 35 \rightarrow C = 2\)- \(3(0) + D = -7 \implies D = -7\)Thus, the constants are \(A = 11\), \(B = 0\), \(C = 2\), and \(D = -7\).

Step 6: Write the Decomposition

The partial fraction decomposition with the determined constants is:\[ \frac{11x + 0}{x^2 + 3} + \frac{2x - 7}{(x^2 + 3)^2} \] Thus, the final decomposition is:\[ \frac{11x}{x^2 + 3} + \frac{2x - 7}{(x^2 + 3)^2} \]

Key Concepts

Rational Expressions

Rational expressions are like fractions, but they contain polynomials in the numerator and the denominator. Just as with regular fractions, we can perform operations like addition, subtraction, multiplication, and division on rational expressions. However, it is important to remember to factor polynomials in the numerator and bottom part of the fraction
to simplify the expression whenever possible.

Partial fraction decomposition is an advanced method that allows us to break down complex rational expressions into a sum of simpler fractions. This is especially useful for integrating or solving large expressions. By splitting the complex fraction into simpler components, it becomes easier to understand and manage each part individually, similar to dividing a large task into smaller, more manageable steps.

In the problem provided, the rational expression is \( \frac{11x^3 + 35x - 7}{x^4 + 6x^2 + 9} \). The goal is to decompose it into a form that reveals more about its behavior and potential solutions, like seeing all the gears inside a clock.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components, or `factors`, that can be multiplied together to get the original polynomial. This is a crucial skill in algebra because many operations, like simplification and solving equations, become significantly easier once the polynomials are factored.

For our exercise, factoring the denominator was the first step in the partial fraction decomposition process. The given denominator was \( x^4 + 6x^2 + 9 \), which initially appears complicated. However, with the right approach, it can be rewritten as \((x^2)^2 + 6(x^2) + 9\) — effectively converting it into a quadratic form. This allows it to be factored as \((x^2 + 3)^2\).

Recognizing patterns, like this quadratic form, is essential for factoring. By practicing different techniques and becoming familiar with patterns, students can identify the most efficient methods to break down complex polynomials into simpler parts. This understanding is fundamental not only in partial fractions but in all areas that involve algebraic expressions.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are polynomials. They work similarly to numeric fractions, but additional complexities arise due to the presence of variables.
This complexity requires additional techniques like factoring and partial fraction decomposition.

A key part of working with algebraic fractions is simplifying them. Simplification often involves canceling common factors in the numerator and denominator or combining like terms. For example, when dealing with the expression \( \frac{11x^3 + 35x - 7}{(x^2 + 3)^2} \), once the denominator is factored, it becomes crucial to apply these techniques to manage each part of the expression.

Understanding how to work with algebraic fractions opens up a wide range of possibilities in mathematics including solving equations and analyzing functions. Once simplified or decomposed, these fractions can play a pivotal role in furthering mathematical understanding and applications.