Decomposing Rational Functions: Mastering Partial Fractions with Repeated Quadratic Factors

PreAlgebra Grades High School 6:08 Video
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Lesson Description

This lesson explores the technique of partial fraction decomposition when dealing with repeated quadratic factors in the denominator. Students will learn how to set up the decomposition, solve for unknown coefficients using strategic substitution and systems of equations, and apply this method to solve complex rational expressions.

Video Resource

Partial Fractions Repeated Quadratic Factors

Mario's Math Tutoring

Duration: 6:08
Watch on YouTube

Key Concepts

  • Partial Fraction Decomposition
  • Repeated Quadratic Factors
  • Strategic Substitution
  • Systems of Equations

Learning Objectives

  • Students will be able to identify rational functions with repeated quadratic factors in the denominator.
  • Students will be able to correctly set up the partial fraction decomposition for rational functions with repeated quadratic factors.
  • Students will be able to solve for the unknown coefficients using strategic substitution and systems of equations.
  • Students will be able to apply partial fraction decomposition to simplify complex rational expressions for integration or other applications.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of partial fraction decomposition and its applications. Briefly discuss the different types of factors that can appear in the denominator (linear, repeated linear, quadratic, repeated quadratic). Introduce the specific case of repeated quadratic factors and explain why they require a different approach.
  • Video Presentation (10 mins)
    Play the video "Partial Fractions Repeated Quadratic Factors" from Mario's Math Tutoring. Encourage students to take notes on the steps involved in setting up the decomposition and solving for the coefficients.
  • Worked Example (15 mins)
    Work through the example presented in the video step-by-step on the board, emphasizing the key steps: 1. Identifying the repeated quadratic factor. 2. Setting up the correct form of the partial fraction decomposition. 3. Clearing the denominators. 4. Using strategic substitution to solve for some coefficients. 5. Setting up and solving a system of equations to find the remaining coefficients. 6. Writing the final partial fraction decomposition.
  • Practice Problems (20 mins)
    Provide students with practice problems involving partial fraction decomposition with repeated quadratic factors. Start with simpler examples and gradually increase the complexity. Encourage students to work in pairs or small groups to discuss their approaches and troubleshoot any difficulties.
  • Wrap-up (5 mins)
    Summarize the key concepts covered in the lesson. Answer any remaining questions. Preview the applications of partial fraction decomposition in calculus (integration of rational functions).

Interactive Exercises

  • Error Analysis
    Present students with incorrect solutions to partial fraction decomposition problems and ask them to identify the errors and explain how to correct them.
  • Group Challenge
    Divide the class into groups and assign each group a challenging partial fraction decomposition problem with repeated quadratic factors. The first group to correctly solve the problem and present their solution to the class wins a small prize.

Discussion Questions

  • Why do we need to include both (x^2 + 1) and (x^2 + 1)^2 in the partial fraction decomposition when we have a repeated quadratic factor of (x^2 + 1)^2?
  • What are the advantages and disadvantages of using strategic substitution versus setting up a system of equations to solve for the unknown coefficients?
  • How does the degree of the numerator in each partial fraction relate to the degree of the corresponding factor in the denominator?

Skills Developed

  • Algebraic Manipulation
  • Problem-Solving
  • Analytical Thinking
  • System of Equations Solving

Multiple Choice Questions

Question 1:

Which of the following denominators indicates the need for partial fraction decomposition with repeated quadratic factors?

Correct Answer: (x^2+1)^2

Question 2:

In the partial fraction decomposition of \(\frac{P(x)}{(x^2+1)^2}\), where P(x) is a polynomial, what is the correct form of the decomposition?

Correct Answer: \(\frac{Ax+B}{x^2+1} + \frac{Cx+D}{(x^2+1)^2}\)

Question 3:

After clearing denominators in a partial fraction decomposition problem, which of the following is a strategic approach to solve for the coefficients?

Correct Answer: Picking convenient values of x

Question 4:

If you have the equation A + B = 5 and A - B = 1, what is the value of A?

Correct Answer: 6

Question 5:

When solving for coefficients using a system of equations, what must be true of the number of equations and unknowns?

Correct Answer: Equal number of equations and unknowns

Question 6:

In the partial fraction decomposition of \(\frac{x^3+1}{(x^2+2)^2}\), what is the degree of the polynomial in the numerator of the partial fraction with denominator (x^2+2)?

Correct Answer: 1

Question 7:

What is the next step after setting up the partial fraction decomposition?

Correct Answer: Clearing the denominators

Question 8:

What method is used to solve for the unknown constants after clearing the denominators?

Correct Answer: Strategic substitution and/or system of equations

Question 9:

What is the benefit of using partial fraction decomposition?

Correct Answer: Simplifies complex rational expressions

Question 10:

If the partial fraction decomposition results in A = 1, B = 2, C = 3, and D = 4, what is the final step?

Correct Answer: Substitute the values back into the partial fraction decomposition setup

Fill in the Blank Questions

Question 1:

A quadratic factor is considered __________ if it appears more than once in the denominator.

Correct Answer: repeated

Question 2:

When decomposing a rational function with a repeated quadratic factor, the numerator of each partial fraction will be one degree __________ than the quadratic factor.

Correct Answer: less

Question 3:

The first step in solving a partial fraction decomposition problem is to __________ the denominators.

Correct Answer: clear

Question 4:

Using strategic _________ of x can simplify the equation and allow you to solve for some of the coefficients.

Correct Answer: substitution

Question 5:

If strategic substitution doesn't solve for all coefficients, a __________ of equations is used to find the remaining variables.

Correct Answer: system

Question 6:

If the repeated quadratic factor is (x^2 + 5)^2, one of the denominators in the partial fraction decomposition is _________.

Correct Answer: (x^2+5)

Question 7:

In general, the number of _________ must match the number of _________ to uniquely determine their values.

Correct Answer: equations

Question 8:

Setting the coefficient of x^2 equal on both sides of the equation generates a new _________ for the system of equations.

Correct Answer: equation

Question 9:

Partial fraction decomposition rewrites complex rational expressions into __________ terms.

Correct Answer: simpler

Question 10:

After solving for the constants A, B, C, D, and E, substitute these values back into the partial fraction __________.

Correct Answer: decomposition

Educational Standards

PC.F.3.3:Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation.

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