MATEMATIK KEJURUTERAAN 4 PARTIAL FRACTIONS B4001

ENGINEERING MATHEMATICS 4 PARTIAL FRACTION BA501

CHAP 4 Partial Fractions

Type of fraction(Proper Fractions Numerator < Denominator Proper fractions have the nominator part smaller than the denominator part, for example INCLUDEPICTURE "http://www.jamit.com.au/images/F1by2.gif" \* MERGEFORMATINET

, or .

(Improper Fractions Numerator > Denominator or Numerator = DenominatorImproper fractions have the nominator part greater or equal to the denominator part, for exampleor.Definition of partial fraction

(Partial fractions use to split a ratio of large polynomials into a sum of ratios of small polynomials. Split two or more fractions from a single fraction.

exmp : =

Types of partial fraction Proper fraction with linear factors( Partial fraction with linear factors is the fraction that has the denominator consists of distinct linear factors ( ax + b )

(in other words do not have any square or cube terms etc).

Example:1 : Find the partial fractions of :

Step:1. Ensure that the degree of the numerator is at least 1 less than the degree of the denominator. 2. Determine the factor of fraction

Condition : i. power of x is 1

ii. power of brackets is 1 3. Apart the fraction by number of factor (the numerator is constant)

= 4. Cross multiplication between partial fraction

= ---------------- equation 15. Use substitution method (kaedah penggantian - anggar nilai x) select the linear factor to be equal to zero

the question have two factors and

i. if

a) Value of constant A can be find because part of A dont has

b) Substitute in equation 1

------ Substitute the value of x

ii. if

a) Value of constant B can be find because part of B dont has

b) Substitute in equation 1

------ Substitute the value of x

6. Replace the constant value in fraction

=

Find the partial fractions of fraction belowi)

2 : Express in partial fractions3 : Convert the fractions below into partial fraction. i)

ii)

Proper fraction with repeated linear factors

Partial fraction with repeated linear factors is the fraction that has the denominator consists of repeated distinct linear factors ( ax + b ) more than one time.

Example: Express the fractions into partial fractions Step:1. Ensure that the degree of the numerator is at least 1 less than the degree of the denominator. 2. Determine the factor of fraction

Condition : i. power of x is 1

ii. power of brackets is more than 1 3. Apart the fraction by number of factor (the numerator is constant)

= 4. Cross multiplication between partial fraction = The factor

---------------- equation 15. Use substitution method (kaedah penggantian - anggar nilai x) select the linear factor to be equal to zero

the question have two factors and

i. if

a) Value of constant A can be find because part of A dont has

b) Substitute in equation 1

---Substitute the value of x

ii. if

a) Value of constant C can be find because part of C dont has

b) Substitute in equation 1

--Substitute the value of x

6. For repeated linear factor, use comparing coefficient ( samakan pekali)

Expand equation 1

Constant B ( has )

Choose (Take only coefficient that have )

7. Replace the constant value in fraction

=

Question:

4 : Express the fractions below into partial fractions i)

ii)

iii)

iv)

Proper fraction with quadratic factorsThis method is for when there is a square term (quadratic factor) in one of the factors of the denominator.

(the quadratic factors which does not factorize.

**every quadratic factors that can factorize, should be solve in other factors.

Example : Solve the fraction .

Step:1. Ensure that the degree of the numerator is at least 1 less than the degree of the denominator. 2. Determine the factor of fraction

Condition : i. power of x is 2

3. Apart the fraction by number of factor (the numerator is constant)

= 4. Cross multiplication between partial fraction = ---------------- equation 15. Use substitution method (kaedah penggantian - anggar nilai x) select the linear factor to be equal to zero

the question has one factors

i. if

a) Value of constant A can be find because part of A dont has

b) Substitute in equation 1

---Substitute the value of x

6. For quadratic factor, use comparing coefficient ( samakan pekali)

Expand equation 1

Constant B ( has )

Choose (Take only coefficient that have )

Constant C ( has )

Choose (Take only coefficient that have )

7. Replace the constant value in fraction

= Question:

5: Solve the fraction below.i)

ii)

iii)

Improper fraction( Polynomial power/degree for numerator is same or more than power for denominator.

( Improper fraction must convert to proper fraction using long division =

Example

6 : Solve the fraction below :

i)

ii)

iii)

iv)

Summaries step to do partial fractionStep

1. Ensure that the degree of the numerator is at least 1 less than the degree of the denominator.

2. Determine the factor of fraction

3. Apart the fraction by number of factor (the numerator is constant)4. Cross multiplication between partial fraction5. For linear factor, use substitution method (kaedah penggantian)

select the factor to be equal to zero

Substitute the value of x

6. For repeated linear factor and quadratic factor, use comparing coefficient ( samakan pekali)

7. Replace the constant value

EMBED Equation.3 = EMBED Equation.3

Exercise 1 :

Express the fractions below into partial fractions:

i) EMBED Equation.3 (ans = EMBED Equation.3

ii) EMBED Equation.3 ( ans = EMBED Equation.3

EMBED Equation.3 = EMBED Equation.3

Exercise 2 :

Express the fractions below into partial fractions:

i) EMBED Equation.3 ( ans = EMBED Equation.3

ii) EMBED Equation.3 ( ans = EMBED Equation.3

EMBED Equation.3 = EMBED Equation.3

Exercise 3 : Convert the fractions below into partial fraction

i) EMBED Equation.3 ( ans = EMBED Equation.3

ii) EMBED Equation.3 ( ans = EMBED Equation.3

Exercise 4 : Solve the fraction below as the partial fraction

i) EMBED Equation.3 ( ans = EMBED Equation.3

ii) EMBED Equation.3 ( ans = EMBED Equation.3

Hint:

To check if you've done the partial fraction expansion correctly, just add all of the partial fractions together to see if their sum equals the original ratio of polynomials.

TUTORIAL

Express the fraction below into partial fraction:

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

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