Lesson Summary(25/1/13) / Partial Fractions summary(Lee Kai En)

Terms

Proper Fraction - A fraction where the numerator < denominator 
Improper Fraction - A fraction where the numerator > denominator
Proper Algebraic Fraction - A fraction where the degree of the numerator < degree of the denominator
Improper Algebraic Fraction - A fraction where the degree of the numerator > degree of denominator

What is partial fraction?
Partial fraction is the breaking up of a fraction to find the original parts of the fraction which will add up to give the final simplified expression.

Methods of finding partial fractions
Method 1: Decomposition
E.G. 

Step 1: Factorise denominator first 
Step 2: Split the fraction up.

Step 3: Make it such that the denominators "disappear"

Methods of solving
Method 1: Comparing coeffs.
E.G.    3x+5 = (2A+B)x + (A-3B)
            2A+B = 3
            A-3B = 5
.......simultaneous equation.......

Method 2: Substitution
E.G      3x+5 ≡ A(2x+1)+B(x-3)
         if x = 3, A = 2
    *(A can be found much easily)
         if x = 

After finding the values of the unknown(such as A and B), substitute in the values into the fractions and express it as a partial fraction.
E.G.

Partial fractions summary (Shaun Ng)

Intro:  Partial fractions

- Instead of simplifying two fractions into one, we want to break them up here.
- Degree is the highest power of x in the set
- Proper fractions = Numerator < Denominator
- Improper fractions = Numerator > Denominator
- Proper algebraic fractions = Degree of numerator < Degree of Denominator
- Improper algebraic fractions = Degree of numerator > Degree of Denominator

Simplifying:

Previously we have been taught how to simplify two fractions into one. (/ sign = border between numerator and denominator)
For example:

8/x+9 + 2/x-3

1st step: Change denominator to the same thing, to do this multiple both numerator and denominator by the denominator in the fraction it is added or subtracted to.

8x-24/(x+9)(x-3) + 2x+18/(x+9)(x-3)

2nd step: Add or subtract the fractions together, by adding the numerators into one fraction.

10x-6/(x+9)(x-3)

This is the answer if we simplify.

Decomposition:

Now lets try something new using the same equation from example of simplifying.

10x-6/(x+9)(x-3)

1st step: Separate into two different fractions with a variable as numerator.

A1/x+9 + A2/x-3

2nd step: Multiply through the bottom to eliminate denominator

A1(x-3) + A2(x+9)

3rd step: Bring the 10x+6 in

10x+6 = A1(x-3) + A2(x+9)

4th step: Find the roots, so that we can eliminate 1 variable and find the other

Root for x-3 is 3
Root for x+9 is -9

5th step: Sub in and solve

Let x be 3
10(3) + 6 = A1(3-3) + A2(3+9)
36 = A2(12)
A2 = 3

Let x be -9
10(-9) + 6 = A1(-9-3) + 3(-9+9)
-96 = A1(-12)
A1 = 8

Hence, A1 = 8 and A2 = 3

6th step: Express as partial fraction

8/x+9 + 3/x-3

PARTIAL FRACTION - summary

BY MR JOHARI

PARTIAL FRACTION 

SUMMARY OF TYPES AS GUIDE

Source: ACE Learning

 

Other Source for reference: http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-partialfractions-2009-1.pdf

LESSON SUMMARY :D:D:D:D:DD:D

Heres what we've covered in class today...

1): Please do remember to organize your workings and provide necessary explanation otherwise the marker would not understand your concept.

2) FRACTIONS: What is a fraction?

A fraction is numerical quantity that is not a whole number 

Numerator/ Denominator

Whole Number- Both values of the numerator and the denominator are equal.

Improper Fraction-  The degree of the unknown in the numerator is higher than the denominator it is said to be a improper fraction 

Proper Fraction– The degree of the unknown in the numerator is lower than the denominator it is said to be a proper fraction.However, if the both the degrees are the same, it would NOT be a proper fraction

3)PARTIAL FRACTIONS

Definition: Partial fraction is breaking apart the final expression into its initial polynomial fractions.

How to solve a Partial Fraction:

Step 1: Factorising the denominator

Step 2: Separate

Step 3: Multiply through by the bottom so there no longer will be fractions

Step 4: Find the constant

Express it as a partial fraction

Summary on Partial fractions ( Mason Sim )

Partial fractions are a way of simplifying fractions that have polynomials in them into simpler , partial fractions , hence the name .  We break fractions into partials fractions so that we can handle them easier with less confusion .

Methods to solve partial fractions 
- Before starting , make sure that the fractions you have are proper ( Numerator degree is smaller than denominator degree. ) If the fraction is improper , use long division or synthetic division in order to get a mixed fraction . In order to make this into mixed fractions , we take the numerator divided by the denominator .

1) Decomposition 
- We factorize the denominator and break it up into factors
-We put different unknowns onto each denominator .
- Therefore , our partial fractions are equal to our original fraction
- Multiply the partial fraction side with the denominator
   >>>
- Start solving it like a normal equation
 ~ You can substitute the x values with those of the denominator such that it will equal 0 such as 2 or -1 in the example shown .



Table

Partial Fractions Summary - Kaelan

Partial Fractions:

Definition: The parts/components of a fraction in proper form of a factorable denominator that when added equal to the original fraction

Methods of solving

Finding the partial fractions (decomposing) of a fraction is the reverse operation of adding its factors.

BUT FIRST

If the degree of numerator is higher than the denominator, take it literally and divide fraction by long division. Then decompose the remainder.

1) Factorise denominator and put equals sign

2) Write a separate fraction for each factor on the RHS as the denominator.
-The numerator is the degree n of the denominator -1
-eg. numerator for linear (x+1) is a constant A -> A/x+1. numerator for quadratic is linear Bx+C...

3) For repeating factors, (x+1)^6. write a separate fraction for  x+1 , (x+1)^2, (x+1)^3 ...

4) Cross multiply numerators on RHS and remove the denominators on both sides.

5) Either compare coefficients or substitute values of x to find the unknowns A,B,C... 

6) Substitute / Rewrite partial fractions with the correct numerator 

                           

Summary on Partial Fractions - Ryan

Partial Fractions: To change an algebraic fraction into 2 separate fractions.

"A rational function which may be expressed as a sum of separate fractions is said to be resolved into its partial fractions."

There are different methods to deal with Proper fractions or Improper fractions

A Proper fraction is a fraction whose numerator is smaller than its denominator.

An Improper fraction is a fraction whose numerator is larger than its denominator.

If the rational function is a proper fraction:
For every linear or quadratic factor there is a corresponding partial fraction.
For every repeated linear factor there are 2 corresponding partial fractions.

To find the unknown constants you can:
compare coefficients
substitution method


If the rational function is an improper fraction:

You must divide it to obtain a quotient and a proper fraction so you can change the proper fraction into partial fraction form.

Partial Fractions- Jemima and Desiree

Partial Fractions

Partial fractions is a way of "breaking apart" fractions with polynomials in them.

Why do we want to do this?
Because the partial fractions are each simpler. This may be a very important step in solving the more complicated fraction.



Partial Fraction Decomposition
Step 1: Factorise the denominator
Step 2: Write one partial fraction for each of the factors
Step 3: Cross multiple so that there are no more fractions
Step 4: Find the constant

It only works for Proper Rational Expressions, where the degree of the top is less than the bottom.

The degree is the largest exponent the variable has.

Type of Fractions:
Proper: the degree of the top is less than the degree of the bottom.
Improper: the degree of the top is greater than, or equal to, the degree of the bottom.


If your expression is improper, then do Polynomial long division first. Obtain the proper fraction (e.g the remainder), then resolve the proper fraction into partial fraction.

Summary on Partial Fractions – Justin

PARTIAL FRACTIONS

Objective: To simplify algebraic fractions into two separate fractions, or a sum of a polynomial and a fraction. 

Types of Fractions:

Not in partial form

Improper– Where the highest power of the unknown in the numerator is higher than that of the denominator. If the powers are the same, the numerator's coefficient of the power with the highest unknown would be higher than that of the denominator.

Proper – Where the highest power of the unknown in the numerator is less than that of the denominator. If both highest powers are equal, the coefficient of the numerator's unknown with the highest power is less than that of the denominator.

In Partial form

'Mixed Fraction' – If the fraction is improper, we must represent it as the sum of a polynomial and a fraction. Thus we have to first use long division, with the denominator as the divisor, to divide the numerator and obtain our polynomial/whole number. 
The remainder would then have to be separated as well. 

Partial Fraction – "Taking apart one algebraic fraction with a quadratic denominator into two separate fractions, often with a linear denominator." 

How to represent fractions as partial fractions:

PARTIAL FRACTION

BY MR JOHARI

A. Introduction to PARTIAL FRACTIONS

RESOURCE: PARTIAL FRACTIONS