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PreAlgebra
Decomposing Rational Functions: Mastering Partial Fractions with Improper Fractions

Decomposing Rational Functions: Mastering Partial Fractions with Improper Fractions

PreAlgebra Grades High School 4:32 Video

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Lesson Description

This lesson delves into the technique of partial fraction decomposition, specifically addressing the scenario when dealing with improper rational functions. It covers identifying improper fractions, using long division to simplify them, and then applying partial fraction decomposition to the resulting proper fraction. This process is crucial for integral calculus and advanced algebraic manipulations.

Video Resource

Partial Fractions with Improper Fractions

Mario's Math Tutoring

Duration: 4:32

Watch on YouTube

Key Concepts

Improper Rational Functions
Long Division of Polynomials
Partial Fraction Decomposition
Linear Factors

Learning Objectives

Identify improper rational functions by comparing the degrees of the numerator and denominator.
Perform long division to convert an improper rational function into the sum of a polynomial and a proper rational function.
Decompose a proper rational function with linear factors into partial fractions.
Solve for the unknown constants in the partial fraction decomposition.
Combine the polynomial term from the long division with the partial fraction decomposition to obtain the complete decomposition of the original improper rational function.

Educator Instructions

Introduction (5 mins)
Begin by reviewing the definition of proper and improper fractions in the context of rational functions. Emphasize that the degree of the numerator must be less than the degree of the denominator for a fraction to be considered proper. Introduce the concept of partial fraction decomposition as a technique to simplify complex rational expressions.
Identifying Improper Fractions (5 mins)
Explain how to identify improper rational fractions. Focus on comparing the highest degree term of both the numerator and denominator. Provide examples of proper and improper rational functions, asking students to identify each type.
Long Division Review (10 mins)
Briefly review the process of long division with polynomials. Work through a simple example to refresh students' memories. Explain that long division is necessary to convert an improper rational function into a form suitable for partial fraction decomposition.
Partial Fraction Decomposition with Improper Fractions (20 mins)
Watch the Mario's Math Tutoring video "Partial Fractions with Improper Fractions." Pause at key points to explain each step in detail. Focus on: * Performing long division on the improper rational function. * Factoring the denominator of the resulting proper rational function. * Setting up the partial fraction decomposition with unknown constants. * Clearing the denominators by multiplying both sides by the common denominator. * Solving for the unknown constants using strategic values of x (as demonstrated in the video). * Writing the final partial fraction decomposition, including the polynomial term from the long division.
Practice Problems (15 mins)
Provide students with practice problems involving partial fraction decomposition of improper rational functions. Encourage them to work independently or in pairs. Circulate to provide assistance and answer questions.
Wrap-up and Q&A (5 mins)
Summarize the key steps involved in partial fraction decomposition of improper rational functions. Answer any remaining questions from students. Assign homework problems for further practice.

Interactive Exercises

Error Analysis
Present students with a worked-out example of partial fraction decomposition with an error. Ask them to identify the mistake and correct it.
Group Problem Solving
Divide students into groups and assign each group a different improper rational function to decompose. Have each group present their solution to the class.

Discussion Questions

Why is it necessary to perform long division before applying partial fraction decomposition to an improper rational function?
What strategies can be used to efficiently solve for the unknown constants in the partial fraction decomposition?
How does the degree of the factors in the denominator influence the form of the partial fraction decomposition?

Skills Developed

Algebraic Manipulation
Problem-Solving
Critical Thinking
Analytical Skills

Multiple Choice Questions

Question 1:

Which of the following rational functions is considered improper?

(x+1)/(x^2+2)

(x^2+3)/(x^3+1)

(x^3+2x)/(x^2+1)

(1)/(x+5)

Correct Answer: (x^3+2x)/(x^2+1)

Question 2:

What is the first step in performing partial fraction decomposition on an improper rational function?

Factor the denominator

Perform long division

Solve for the constants

Set up the partial fractions

Correct Answer: Perform long division

Question 3:

After performing long division on an improper rational function, you obtain a quotient and a remainder. What do you do with the remainder?

Ignore the remainder

Decompose the remainder over the original divisor

Add the remainder to the quotient

Integrate the remainder

Correct Answer: Add the remainder to the quotient

Question 4:

In partial fraction decomposition, what is the correct form for a linear, non-repeated factor (x+a) in the denominator?

A/(x+a)

A/(x+a)^2

A/(x+a) + B/(x+a)^2

Ax + B

Correct Answer: A/(x+a)

Question 5:

When solving for the constants in partial fraction decomposition, what is a strategic value to substitute for x?

x = 0

x = 1

x = -a, where (x+a) is a factor of the denominator

x = any random number

Correct Answer: x = -a, where (x+a) is a factor of the denominator

Question 6:

What is the purpose of partial fraction decomposition?

To simplify rational functions into simpler fractions

To make rational functions more complicated

To solve differential equations

To find the derivative of a rational function

Correct Answer: To simplify rational functions into simpler fractions

Question 7:

Which of the following is a proper rational function?

(x^2+1)/(x+3)

(x^4)/(x^2+1)

(5)/(x^3+2x+1)

(x^3+2)/(x^3-1)

Correct Answer: (5)/(x^3+2x+1)

Question 8:

If you have the expression (5x+1)/(x+1)(x-2), what will the setup of your partial fraction decomposition look like?

A/(x+1)

A/(x+1) + B/(x-2)

A/(x+1) + B/(x-2) + C

A/(x+1)^2 + B/(x-2)

Correct Answer: A/(x+1) + B/(x-2)

Question 9:

After decomposing (4x+7)/(x+2)(x+1) into partial fractions, you obtain A/(x+2) + B/(x+1). What are the values of A and B?

A = 1, B = 3

A = 3, B = 1

A = -1, B = 3

A = -3, B = 1

Correct Answer: A = 1, B = 3

Question 10:

Given an improper rational function, you've performed long division and obtained x + (4x+7)/(x^2+3x+2). What is the next step in fully decomposing this expression?

Factor x^2+3x+2 and perform partial fraction decomposition on (4x+7)/(x^2+3x+2)

Add x to the numerator of (4x+7)/(x^2+3x+2)

Take the derivative of the whole expression

Simplify the expression by combining like terms

Correct Answer: Factor x^2+3x+2 and perform partial fraction decomposition on (4x+7)/(x^2+3x+2)

Fill in the Blank Questions

Question 1:

A rational function is considered __________ if the degree of the numerator is greater than or equal to the degree of the denominator.

Correct Answer: improper

Question 2:

Before performing partial fraction decomposition on an improper rational function, we must use __________ to rewrite it.

Correct Answer: long division

Question 3:

When decomposing a rational function with a linear factor of (x-a) in the denominator, the corresponding partial fraction will have the form A/__________.

Correct Answer: (x-a)

Question 4:

To clear the denominators when solving for the constants in partial fraction decomposition, multiply both sides of the equation by the __________.

Correct Answer: common denominator

Question 5:

In the context of partial fractions, the reverse process of combining partial fractions back into a single fraction is called __________.

Correct Answer: decomposition

Question 6:

The terms A and B used in partial fraction decomposition represent __________.

Correct Answer: constants

Question 7:

After long division of an improper fraction, the fraction containing the remainder is considered a __________ fraction.

Correct Answer: proper

Question 8:

The factors in the denominator of the proper fraction will always be __________ to be properly solved using partial fraction decomposition.

Correct Answer: linear

Question 9:

When solving for A and B, we can substitute x values to strategically zero out different parts of our equation, these values are typically the __________ of our factors.

Correct Answer: zeroes

Question 10:

After solving for our A and B values, we must remember to include the __________ from the long division problem as part of our final answer.

Correct Answer: quotient

Educational Standards

PC.F.1:Analyze functions and relations. PC.F.1.4:Describe end behavior, asymptotic behavior, and points of discontinuity. PC.F.3.1:Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic. PC.F.3.3:Algebraically verify solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

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