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The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

Partial Fractions

Samdom For Peace

I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.
The Wolfram Alpha widgets (many thanks to the developers) were used for some calculators.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

Objectives

Objectives

Students will:

(1.) Discuss algebraic fractions.

(2.) Decompose/Resolve whole algebraic fractions into partial algebraic fractions.

(3.) Compose/Add partial algebraic fractions into a whole algebraic fraction.


Skills Measured/Acquired

(1.) Use of prior knowledge

(2.) Critical Thinking

(3.) Interdisciplinary connections/applications

(4.) Technology

(5.) Active participation through direct questioning

(6.) Research

Vocabulary Words

Please note:
added to
subtracted from
multiplied by
divided by

Check for prior knowledge. Ask students about these terms.

Bring it to English: partial, total, whole, complete, vary, constant, express, expression, equate, equal, equation, equality, equanimity, equity, addendum

Bring it to Math: arithmetic, arithmetic operators, sum, difference, product, quotient, augend, addend, minuend, subtrahend, multiplier, multiplicand, factor, dividend, divisor, positive, negative, nonpositive, nonnegative, constant, number, variable, term, add, subtract, multiply, divide, expression, equation, equal, linear, quadratic, cubic, quartic, exponent, index, power, degree, order, etc.

Definitions

A fraction is a part of a whole
It is the part of something out of a whole thing.
It is also seen as a ratio.
It is also seen as a quotient.

The numerator is the part.
It is the "top" part of the fraction.

The denominator is the whole.
It is the "bottom" part of the fraction.

A Proper Fraction is a fraction whose numerator is less than the denominator.
A Proper Algebraic Fraction is a fraction whose degree of the numerator is less than the degree of the denominator.

An Improper Fraction is a fraction whose numerator is greater than or equal to the denominator.
An Improper Algebraic Fraction is a fraction whose degree of the numerator is greater than or equal to the degree of the denominator.

Equivalent Fractions are two or more fractions that have the same value when they are expressed in their simplest forms.

Common Denominators are the common multiples of the different denominators of unlike fractions.

Least Common Denominator is the least of all the common multiples of the different denominators of unlike fractions.

Prime Factorization is a method used for finding the least common denominator of unlike fractions, in which each denominator is broken down into a product of prime numbers. This means that each denominator is split into a product of prime factors.

The addition of Partial Fractions gives a Whole Fraction (Sum)

A ratio is a comparison of two quantities.
A ratio is seen as a fraction.

A percent means something out of $100$.
A percent is also seen as a fraction.

The basic arithmetic operators are the addition symbol, $+$, the subtraction symbol, $-$, the multiplication symbol, $*$, and the division symbol, $\div$

Augend is the term that is being added to. It is the first term.

Addend is the term that is added. It is the second term.

Sum is the result of the addition.

$$3 + 7 = 10$$ $$3 = augend$$ $$7 = addend$$ $$10 = sum$$

Minuend is the term that is being subtracted from. It is the first term.

Subtrahend is the term that is subtracted. It is the second term.

Difference is the result of the subtraction.

$$3 - 7 = -4$$ $$3 = minuend$$ $$7 = subtrahend$$ $$-4 = difference$$

Multiplier is the term that is multiplied by. It is the first term.

Multiplicand is the term that is multiplied. It is the second term.

Product is the result of the multiplication.

$$3 * 10 = 30$$ $$3 = multiplier$$ $$10 = multiplicand$$ $$30 = product$$

Dividend is the term that is being divided. It is the numerator.

Divisor is the term that is dividing. It is the denominator.

Quotient is the result of the division.

Remainder is the term remaining after the division.

$$12 \div 7 = 1 \:R\: 5$$ $$12 = dividend$$ $$10 = divisor$$ $$1 = quotient$$ $$5 = remainder$$

A constant is something that does not change. In mathematics, numbers are usually the constants.

A variable is something that varies (changes). In Mathematics, alphabets are usually the variables.

A mathematical expression is a combination of variables and/or constants using arithmetic operators.

A linear expression is an expression in which the highest exponent of the independent variable in the expression is $1$

A quadratic expression is an expression in which the highest exponent of the independent variable in the expression is $2$

A cubic expression is an expression in which the highest exponent of the independent variable in the expression is $3$

A quartic expression is an expression in which the highest exponent of the independent variable in the expression is $4$

Introduction

Please ensure you have reviewed the Prerequisite Topics: Fractions and Factoring

Let us begin with Arithmetic

$ \dfrac{2}{3} + \dfrac{5}{8} \\[5ex] = \dfrac{16}{24} + \dfrac{15}{24} \\[5ex] = \dfrac{16 + 15}{24} \\[5ex] = \dfrac{31}{24} \\[5ex] $ $\dfrac{2}{3}$ and $\dfrac{5}{8}$ are the partial fractions

$\dfrac{31}{24}$ is the whole fraction

It is an improper fraction

$ \dfrac{1}{2} + \dfrac{1}{5} + \dfrac{1}{8} \\[5ex] = \dfrac{20}{40} + \dfrac{8}{40} + \dfrac{5}{40} \\[5ex] = \dfrac{20 + 8 + 5}{40} \\[5ex] = \dfrac{33}{40} \\[5ex] $ $\dfrac{1}{2}$, $\dfrac{1}{5}$, and $\dfrac{1}{8}$ are the partial fractions

$\dfrac{33}{40}$ is the whole fraction

It is a proper fraction

Ask students if they can work the other way around -

decompose $\dfrac{31}{24}$ into $\dfrac{2}{3}$ and $\dfrac{5}{8}$?

decompose $\dfrac{33}{40}$ into $\dfrac{1}{2}$, $\dfrac{1}{5}$, and $\dfrac{1}{8}$?

Note their responses and respond accordingly
Notice that some of them may give different correct answers

What about Algebra?

$ \dfrac{2x}{3} + \dfrac{5x}{8} \\[5ex] = \dfrac{16x}{24} + \dfrac{15x}{24} \\[5ex] = \dfrac{16x + 15x}{24}x \\[5ex] = \dfrac{31x}{24} \\[5ex] $ $\dfrac{2x}{3}$ and $\dfrac{5x}{8}$ are the partial algebraic fractions

$\dfrac{31x}{24}$ is the whole algebraic fraction

It is an improper fraction

$ \dfrac{1x}{2} + \dfrac{1x}{5} + \dfrac{1x}{8} \\[5ex] = \dfrac{20x}{40} + \dfrac{8x}{40} + \dfrac{5x}{40} \\[5ex] = \dfrac{20x + 8x + 5x}{40} \\[5ex] = \dfrac{33x}{40} \\[5ex] $ $\dfrac{1x}{2}$, $\dfrac{1x}{5}$, and $\dfrac{1x}{8}$ are the partial fractions

$\dfrac{33x}{40}$ is the whole fraction

It is a proper fraction

Ask students if they can work the other way around -

decompose $\dfrac{31x}{24}$ into $\dfrac{2x}{3}$ and $\dfrac{5x}{8}$?

decompose $\dfrac{33x}{40}$ into $\dfrac{1x}{2}$, $\dfrac{1x}{5}$, and $\dfrac{1x}{8}$?

Note their responses and respond accordingly
Notice that some of them may give different correct answers

In Mathematics, we do not really want to have different correct answers
We can have different ways/methods of solving the same question to get the correct answer
However, we want to have only one correct answer rather than different answers
We want to have specific partial fractions rather than several partial fractions

Student: Why do we need to work the other way around in the first place?
Teacher: Would it not be interesting to disassemble all the parts of a computer just to see all the parts that are inside?
Student: Why would you want to do that?
Teacher: Assume the computer stops working because of a hardware
You would want to find out the part(s) that are damaged so you can repair or replace it
Disassembling a computer into several parts is similar to decomposing a whole fraction into partial fractions
Another reason: This topic will lead us to the next topic: Consequent Topic: Integration by Partial Fractions

So, we are going to deal with those whole algebraic fractions that will give us specific partial fractions.
We shall look at all the forms of whole algebraic fractions - both the whole proper fractions and the whole improper fractions

General Techniques of Decomposing Whole Fractions into Partial Fractions

For Proper Fractions (Degree of Numerator is less than Degree of Denominator)
Simplify all the factors at the denominator if possible to determine the kind of factors
Watch out for repeated factors, and recognize each factor up to the number of times it was repeated
(1.) If the denominator has a linear factor, the corresponding numerator should be a constant

$ (a.)\:\: \dfrac{x - 1}{(x + 3)(2x -7)} = \dfrac{A}{x + 3} + \dfrac{B}{2x - 7} \\[5ex] (b.)\:\: \dfrac{x - 1}{2x^2 - x - 21} = \dfrac{x - 1}{(x + 3)(2x - 7)} = \dfrac{A}{x + 3} + \dfrac{B}{2x - 7} \\[7ex] $ (2.) If the denominator has a quadratic factor, the corresponding numerator should be a linear factor

$ \dfrac{3 - 2x}{(x^2 + 7)(x^2 - x + 1)} = \dfrac{Ax + B}{x^2 + 7} + \dfrac{Cx + D}{x^2 - x + 1} \\[7ex] $ (3.) If the denominator has a cubic factor, the corresponding numerator should be a quadratic factor

$ \dfrac{4}{(5 - 2p^3)(3p^3 - 2p^2 + 7p - 3)} = \dfrac{Ap^2 + Bp + C}{5 - 2p^3} + \dfrac{Dp^2 + Ep + F}{3p^3 - 2p^2 + 7p - 3} \\[7ex] $ (4.) If the denominator has a mixture of a linear factor, quadratic factor, and cubic factor; the corresponding numerators should be a constant, a linear factor, and a quadratic factor

$ \dfrac{3 - 2x}{x(2x^2 + 3x - 7)(3x^3 - 5x^2 + 12)} = \dfrac{A}{x} + \dfrac{Bx + C}{2x^2 + 3x - 7} + \dfrac{Dx^2 + Ex + F}{3x^3 - 5x^2 + 12} \\[7ex] $ (5.) If the denominator has a repeated linear factor; the corresponding numerators should be a constant for the first linear factor, and another constant for the repeated linear factor

$ \dfrac{3p}{(p - 9)(p - 9)} = \dfrac{3p}{(p - 9)^2} = \dfrac{A}{p - 9} + \dfrac{B}{(p - 9)^2} \\[7ex] $ (6.) If the denominator has a repeated non-linear (quadratic) factor; the corresponding numerators should be a linear factor for the first quadratic factor, and another linear factor for the repeated quadratic factor

$ \dfrac{3 - 2x + 7x^2}{(7x^2 - 5x - 11)^2} = \dfrac{Ax + B}{7x^2 - 5x - 11} + \dfrac{Cx + D}{(7x^2 - 5x - 11)^2} \\[7ex] $ (7.) If the denominator has a repeated/non-repeated linear/non-linear factors; write the numerators accordingly

$ \dfrac{x^2 + 5}{x(x^3 + x} = \dfrac{x^2 + 5}{x(x^2 + 1)} = \dfrac{A}{x} + \dfrac{Bx + C}{x^2 + 1} \\[7ex] $ For Improper Fractions (Degree of Numerator is equal to or greater than Degree of Denominator)

Please see more examples of these forms in their respective sections.

Proper Fractions: Non-Repeated Linear Factors at the Denominator

Please ensure you have reviewed the Prerequisite Topic: Factoring first.

This form:
(1.) is a proper algebraic fraction (the degree of the numerator is less than the degree of the denominator)
(2.) has linear factors at the denominator
(3.) has factor(s) at the denominator that can be simplified as linear factors.
This implies that you have to simplify any factor(s) at the denominator whenever possible before you determine the form.

Some examples of this form are:

$ (a.)\:\: \dfrac{x - 1}{(x + 3)(2x -7)} = \dfrac{A}{x + 3} + \dfrac{B}{2x - 7} \\[5ex] (b.)\:\: \dfrac{x - 1}{2x^2 - x - 21} = \dfrac{x - 1}{(x + 3)(2x - 7)} = \dfrac{A}{x + 3} + \dfrac{B}{2x - 7} \\[5ex] (c.)\:\: \dfrac{7}{x(2x - 1)(3x + 2)} = \dfrac{A}{x} + \dfrac{B}{2x - 1} + \dfrac{C}{3x + 2} \\[5ex] (d.)\:\: \dfrac{6 - 2p}{9p^2 - 16} = \dfrac{6 - 2p}{3^2p^2 - 4^2} = \dfrac{6 - 2p}{(3p)^2 - 4^2} = \dfrac{6 - 2p}{(3p + 4)(3p - 4)} = \dfrac{A}{3p + 4} + \dfrac{B}{3p - 4} \\[5ex] (e.)\:\: \dfrac{3k}{(k - 1)(2 - k)(4 + k)(k - 5)} = \dfrac{A}{k - 1} + \dfrac{B}{2 - k} + \dfrac{C}{4 + k} + \dfrac{D}{k - 5} $

Proper Fractions: Non-Repeated Non-Linear Factors at the Denominator

Please ensure you have reviewed the Prerequisite Topic: Factoring first.

This form:
(1.) is a proper algebraic fraction (the degree of the numerator is less than the degree of the denominator)
(2.) has non-linear factors (quadratic factors, cubic factors, quartic factors, etc) at the denominator
(3.) has factor(s) at the denominator that cannot be simplified as linear factors.
This implies that you have to simplify any factor(s) at the denominator whenever possible before you determine the form.

Some examples of this form are:

$ (a.)\:\: \dfrac{3 - 2x}{(x^2 + 7)(x^2 - x + 1)} = \dfrac{Ax + B}{x^2 + 7} + \dfrac{Cx + D}{x^2 - x + 1} \\[5ex] (b.)\:\: \dfrac{4}{(5 - 2p^3)(3p^2 - 2p + 7)} = \dfrac{4}{(3p^2 - 2p + 7)(5 - 2p^3)} = \dfrac{Ap + B}{3p^2 - 2p + 7} + \dfrac{Cp^2 + Dp + E}{5 - 2p^3} \\[5ex] OR \\[3ex] (b.)\:\: \dfrac{4}{(5 - 2p^3)(3p^2 - 2p + 7)} = \dfrac{Ap^2 + Bp + C}{5 - 2p^3} + \dfrac{Dp + E}{3p^2 - 2p + 7} \\[5ex] (c.)\:\: \dfrac{4x^2 - 3x + 10}{(3x^4 - 5)(4 - 7x^3)} = \dfrac{Ax^3 + Bx^2 + Cx + D}{3x^4 - 5} + \dfrac{Ex^2 + Fx + G}{4 - 7x^3} \\[5ex] OR \\[3ex] (c.)\:\: \dfrac{4x^2 - 3x + 10}{(3x^4 - 5)(4 - 7x^3)} = \dfrac{4x^2 - 3x + 10}{(4 - 7x^3)(3x^4 - 5)} = \dfrac{Ax^2 + Bx + C}{4 - 7x^3} + \dfrac{Dx^3 + Ex^2 + Fx + G}{3x^4 - 5} $

Proper Fractions: Repeated Linear Factors at the Denominator

Please ensure you have reviewed the Prerequisite Topic: Factoring first.

This form:
(1.) is a proper algebraic fraction (the degree of the numerator is less than the degree of the denominator)
(2.) has repeated linear factors at the denominator
(3.) has factor(s) at the denominator that can be simplified as repeated linear factors.
This implies that you have to simplify any factor(s) at the denominator whenever possible before you determine the form.

Some examples of this form are:

$ (a.)\:\: \dfrac{-3x}{(x + 4)^2} = \dfrac{A}{x + 4} + \dfrac{B}{(x + 4)^2} \\[5ex] (b.)\:\: \dfrac{3p}{(p - 9)(p - 9)} = \dfrac{3p}{(p - 9)^2} = \dfrac{A}{p - 9} + \dfrac{B}{(p - 9)^2} \\[5ex] (c.)\:\: \dfrac{7 - 2k}{k^2(2x - 3)^2} = \dfrac{A}{k} + \dfrac{B}{k^2} + \dfrac{C}{2k - 3} + \dfrac{D}{(2k - 3)^2} \\[5ex] (d.)\:\: \dfrac{12}{x^3(2 - 3x)^3} = \dfrac{A}{x} + \dfrac{B}{x^2} + \dfrac{C}{x^3} + \dfrac{D}{2 - 3x} + \dfrac{E}{(2 - 3x)^2} + \dfrac{F}{(2 - 3x)^3} $

Proper Fractions: Repeated Non-Linear Factors at the Denominator

Please ensure you have reviewed the Prerequisite Topic: Factoring first.

This form:
(1.) is a proper algebraic fraction (the degree of the numerator is less than the degree of the denominator)
(2.) has repeated non-linear factors at the denominator
(3.) has factor(s) at the denominator that can be simplified as repeated non-linear factors.
This implies that you have to simplify any factor(s) at the denominator whenever possible before you determine the form.

Some examples of this form are:

$ (a.)\:\: \dfrac{3 - 2x + 7x^2}{(7x^2 - 5x - 11)^2} = \dfrac{Ax + B}{7x^2 - 5x - 11} + \dfrac{Cx + D}{(7x^2 - 5x - 11)^2} \\[5ex] (b.)\:\: \dfrac{5p^2 + 7}{(3p^2 + 7)^3(2p^3 - 5)^2} = \dfrac{Ap + B}{3p^2 + 7} + \dfrac{Cp + D}{(3p^2 + 7)^2} + \dfrac{Ep + F}{(3p^2 + 7)^3} + \dfrac{Gp^2 + Hp + I}{2p^3 - 5} + \dfrac{Jp^2 + Kp + L}{(2p^3 - 5)^2} $

Proper Fractions: Repeated/Non-repeated Linear/Non-Linear Factors at the Denominator

Please ensure you have reviewed the Prerequisite Topic: Factoring first.

This form:
(1.) is a proper algebraic fraction (the degree of the numerator is less than the degree of the denominator)
(2.) has repeated/non-repeated linear/non-linear factors at the denominator
Please make sure you simplify any factor(s) at the denominator whenever possible before you determine the form.

Some examples of this form are:

$ (a.)\:\: \dfrac{89}{x^2 - 6x} = \dfrac{89}{x(x - 6)} = \dfrac{A}{x} + \dfrac{B}{x - 6} \\[5ex] (b.)\:\: \dfrac{x^2 + 108x + 108}{x^3 - 4x} = \dfrac{x^2 + 108x + 108}{x(x^2 - 4)} = \dfrac{x^2 + 108x + 108}{x(x^2 - 2^2)} = \dfrac{x^2 + 108x + 108}{x(x + 2)(x - 2)} = \dfrac{A}{x} + \dfrac{B}{x + 2} + \dfrac{C}{x - 2} \\[5ex] (c.)\:\: \dfrac{11x^2 + 2x - 8}{x^3 + x^2} = \dfrac{11x^2 + 2x - 8}{x^2(x + 1)} = \dfrac{A}{x} + \dfrac{B}{x^2} + \dfrac{C}{x + 1} \\[5ex] (d.)\:\: \dfrac{x^2 + 5}{x(x^3 + x} = \dfrac{x^2 + 5}{x(x^2 + 1)} = \dfrac{A}{x} + \dfrac{Bx + C}{x^2 + 1} \\[5ex] $

Improper Fractions

Please ensure you have reviewed the Prerequisite Topics: Synthetic Division and Long Division

This form:
(1.) is an improper algebraic fraction (the degree of the numerator is equal to or greater than the degree of the denominator)
You have to use the Long Division or the Synthetic Division to convert the improper fraction to a mixed number (an integer and a proper fraction)

Some examples of this form are: