Examples on Partial Fractions

Examples on Partial Fractions Recall the following 3 types of partial fractions: (if the fraction is a “proper” fraction that is) Type 1:



Type 2(i):



Type 2(ii):



Type 3:



Cover-up Rule can be used completely for this Cover-up Rule can be used to find this only!

(where and

Important Note:

Example 1: Express

are positive)

in partial fractions. The denominator is NOT linear and also NOT “Type 3” linear, so we must factorize the denominator.

Solution:

Remember to factorize the denominator if it is not linear and NOT “Type 3”.

Important Note on factorization: Most people use the calculator (mode 2,2) to help them factorize. But care must be taken to ensure that our factorization is correct (due to the coefficient of the - term not being “1”):

Applying partial fractions:

From using “mode 2,2”, we get: “ “

” and

” . So our factorization would be: . But that isn’t the correct factorization.

Using “Cover-up Rule”: Here, we let

Since the coefficient of the -term in is “2”, we must multiply the factorization obtained from the calculator by “2”. Thus the correct factorization would be:

Here, we let

. The

“2” can be multiplied into the factor with the fraction, giving:

Example 2: Express

in partial fractions.

Factorize this as it is a Quadratic Expression that is NOT “Type 3”

Solution: Factorize the denominator This is Type 2(ii)

By Cover-up Rule:

Only these to 2 can be found by “cover-up” Rule Here, we let

Here, we let

To find : We let Thus we let

be any value that is not

or “2”

:

in

Solve for

Thus:

As we have use these in “Coverup”, We cant use or .

Example 3: Express

in partial fractions. This is a Quadratic Expression of “Type 3”. So we shall apply “Type 3” Partial Fraction

Solution: Before we apply “Type 3” Partial Fractions, Realize that the fraction is not a proper fraction as the degree of the numerator (degree of numerator ) is the same as the degree of the denominator (Degree of numerator ). Thus, we shall apply Long Division first. To apply long division first, we must ensure both the numerator & denominator are in expanded form and write the numerator & denominator in descending powers:

Write

in descending powers of :

This is the quotient

Stop the long division when the degree of the remainder is smaller than the degree of divisor

After long division, the division can be written as: Apply Partial Fractions on this

“Type 3” Partial Fractions

Have to factorize denominator to in order to apply Partial Fractions

Apply “Cover-up” Rule on this!

By Cover-up Rule:

To find and , one have to multiply the denominator on to get rid of ALL fractions and then COMPARE COEFFICIENTS on both sides of the equation

on both sides

Collate the same terms together on

terms

terms

terms

By comparing coefficients of the same terms on both sides: Comparing

-term on both sides:

Comparing constant term on both sides:

Thus:

This is the Answer