This is the 3rd pattern on the topic of partial fraction.
The topic of partial fraction is quite simple but there are still some tricks that the "O" level examiners used in setting questions on partial fractions.
Perhaps you will like to do these questions.
Question 1
Express x^2/(9-x^2) in the form A + B/(3-x) + C/(3+x)
and find the values of A, B and C.
[The trick on this question is on the constant A]
Question 2
Express in partial fractions
(2x^3 - 3x^2 - 7x + 4)/[(x-1)(x+1)]
I have used the long-division and cover up method to solve the questions.
However, these two questions can also be solved by using the comparison method or substitution method ie with the use of long division for example 2 too.
Question 1
Express x^2/(9-x^2) in the form A + B/(3-x) + C/(3+x)
and find the values of A, B and C.
[The trick on this question is on the constant A]
Answer
Use Long Division,
x^2/(9-x^2) = -1 + 9/(x^2-1)
= -1 + 9/[(x-3)(x+3)] = A + B/(3-x) + C/(3+x)
Use Cover up method,
= -1 + 1.5/(3-x) + 1.5/(3+x)
Hence, A = -1, B = 1.5 and C = 1.5
Question 2
Express in partial fractions
(2x^3 - 3x^2 - 7x + 4)/[(x-1)(x+1)]
Answer
(2x^3 - 3x^2 - 7x + 4)/[(x-1)(x+1)]
= (2x^3 - 3x^2 - 7x + 4)/(x^2-1)
Use long Division,
= 2x - 3 + (1-5x)/(x^2 -1)
= 2x - 3 + (1-5x)/[(x -1)(x+1)] = 2x-3 + A/(x -1) + B/(x+1)
= 2x -3 -2/(x-1) – 3/(x+1)
Thank you for your kind attention.
Regards
ahm97sic
PS : The objective of Example 1 is to let students to know/realise that it is not correct to
express in partial fractions as
x^2/9-x^2 = x^2/[(3-x)(3+x)] = A/(3-x) + B/(x+3)
it should be
x^2/(9-x^2) = -1 + 9/(x^2-1) = -1 + 9/[(x-3)(x+3)] = A + B/(3-x) and C/(3+x)
The objective of example 2 is let the students to know / realize the trick involved when they use the long division, there is a trick involved. You must do the long division for this question to realise the trick.