Problem 17 - Algebra for College Students

Question

Find the partial fraction decomposition for each rational expression. See answers below. \(\frac{3 x^{2}+10 x+9}{(x+2)^{3}}\)

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{3}{x+2} - \frac{2}{(x+2)^2} + \frac{1}{(x+2)^3}\).

Step 1: Identify the Denominator

We have the rational expression \(\frac{3x^{2}+10x+9}{(x+2)^{3}}\). The denominator \((x+2)^3\) suggests that the partial fraction decomposition will involve terms with \(x+2\) raised to the powers 1, 2, and 3.

Step 2: Set Up the Partial Fraction Decomposition

Write the expression as a sum of partial fractions. For the factors \((x+2)\), \((x+2)^2\), and \((x+2)^3\), the decomposition is:\[\frac{3x^{2}+10x+9}{(x+2)^{3}} = \frac{A}{x+2} + \frac{B}{(x+2)^{2}} + \frac{C}{(x+2)^{3}}\] where \(A\), \(B\), and \(C\) are constants to be determined.

Step 3: Clear the Denominator

Multiply both sides of the equation by \((x+2)^{3}\) to eliminate fractions:\[3x^2 + 10x + 9 = A(x+2)^{2} + B(x+2) + C\] This produces a polynomial equation where the right side needs to be expanded.

Step 4: Expand and Combine Like Terms

Expand the right side of the equation:\[ A(x+2)^2 = A(x^2 + 4x + 4) = Ax^2 + 4Ax + 4A \]\[ B(x+2) = Bx + 2B \]Now combine all terms on the right:\[Ax^2 + 4Ax + 4A + Bx + 2B + C\] Group terms with similar powers of \(x\):\[Ax^2 + (4A + B)x + (4A + 2B + C)\]

Step 5: Compare Coefficients

Match the coefficients of corresponding powers of \(x\) from both sides of the equation:- For \(x^2\): \(A = 3\)- For \(x^1\): \(4A + B = 10\)- For the constant term: \(4A + 2B + C = 9\)

Step 6: Solve the System of Equations

From \(A = 3\), substitute into the other equations:- \(4(3) + B = 10 \Rightarrow 12 + B = 10 \Rightarrow B = -2\)- Substitute \(A\) and \(B\) into the constant term equation: \[4(3) + 2(-2) + C = 9 \Rightarrow 12 - 4 + C = 9 \Rightarrow C = 1\]

Step 7: Write the Final Decomposition

Incorporate the solved values \(A = 3\), \(B = -2\), \(C = 1\) into the partial fraction decomposition:\[\frac{3}{x+2} - \frac{2}{(x+2)^2} + \frac{1}{(x+2)^3}\] This is the final partial fraction decomposition.

Key Concepts

Understanding Rational Expressions

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. For instance, in the partial fraction decomposition exercise, you were given \(\frac{3x^2 + 10x + 9}{(x+2)^3}\). This represents a rational expression with a polynomial in the numerator, \(3x^2 + 10x + 9\), and a polynomial in the denominator, \((x + 2)^3\).
These expressions are common in algebra and calculus because they help describe rational functions, which are essential for modeling real-world phenomena.
To work with rational expressions, understanding each component is crucial:

• Numerator: The top part of the fraction, which is a polynomial equation.
• Denominator: The bottom part that must not be zero, also a polynomial.
• Significance: Simplifying rational expressions can help in solving equations, performing integration, or even graphing functions.

By learning to decompose these expressions into sums of simpler fractions, like in the given solution, we make them easier to work with.

Characteristics of Polynomial Equations

Polynomial equations form the backbone of many mathematical expressions and calculations. A polynomial equation consists of terms with variables raised to whole number powers and has coefficients.
In the exercise's context, when dealing with \(3x^2 + 10x + 9\), we focus on identifying polynomial characteristics. This involves determining degrees, which is the highest power an equation is raised to, and understanding the role of each term's coefficients.
Let's break it down:

• Degree: The degree of a polynomial is crucial for understanding its potential behavior. In the numerator \(3x^2 + 10x + 9\), the degree is 2.
• Coefficients: These are the numbers in front of the variable terms, such as 3 in \(3x^2\). Coefficients affect the steepness, direction, and crossing points of curves on a graph.
• Innovations: Whether solving manually or using technology, recognizing polynomial equations’ characteristics helps in expanding, factoring, and solving these essential algebraic expressions.

Understanding polynomial equations can reveal the roots of functions, critical points for graphing, and much more.

Applying Systems of Equations

Systems of equations arise frequently in mathematics, especially in the analysis of partial fraction decomposition.
These systems involve finding the values of variables that satisfy multiple equations at once. For example, after clearing the denominators and setting up the equation from the decomposition, we have to solve:

• \( A = 3 \)
• \( 4A + B = 10 \)
• \( 4A + 2B + C = 9 \)

• Utility: Systems of equations help to find unknowns, such as the constants \(A\), \(B\), and \(C\) in our example, that make the initial polynomial equation equivalent on both sides.
• Simplification: Solving these systems involves techniques like substitution or elimination, which require mathematical precision and logic.
• Flexibility: Experienced mathematicians often develop shortcuts or use software to quickly arrive at solutions — an essential strategy for more complex systems.

Recognizing these systems’ connections in solving mathematical problems prepares students for advanced algebra and calculus topics.

Mathematics Education and Learning Methods

In the realm of mathematics education, the relevance of techniques like partial fraction decomposition becomes clear in building foundational skills for students.
New mathematical topics introduce methods and strategies that build on prior knowledge and expand understanding in a structured way. This process aids learners as they work through increasingly complex problems, just like the rational expressions exercise.
Here are some key learning methods to bolster understanding:

• Conceptual Understanding: Before delving into calculations, grasp the underlying concept. Know what partial fractions represent and why they're useful.
• Practice and Repetition: Frequent exercise with different rational expressions enhances both confidence and competence.
• Step-by-Step Problem Solving: Break down tasks into manageable steps, as shown in the original exercise solution. It’s a reliable way to avoid mistakes and reinforce learning.
• Peer Discussions and Collaboration: Working in groups helps see different perspectives and methodologies, enriching the learning experience.

Through a thoughtful approach to teaching and learning, students enhance their analytical skills and better appreciate the interconnectedness of various mathematical concepts.