Cover up Rule in Partial Fraction

Introduction The cover up rule is a
faster technique in finding constants in partial fraction. We assume that the
reader already knows some elementary methods of breaking a rational function
into its appropriate partial fraction. We can only apply the
cover up rule when the denominator is a product of linear factors. 
Example 1 Consider the partial
fraction To obtain A simply cover up the factor (x –1) with you finger tip in
then
you get : and
substitute the value x = 1, giving Likewise to obtain B cover up the factor
(x + 2) in (1) and
evaluate what is left at x = 2, giving Finally
you get: 
Example 2 If there are three
factors, we can find the corresponding constants just by covering up each
factor in the denominator one by one and substitute the root of the linear
factor covered in the remaining fraction. The reader may check the
following (don’t press the screen too hard if you are using LCD monitor!) : Then: Therefore: 
Example 3 Suppose the linear
factors in the denominator are not linear, the cover up rule can still be
helpful. Consider: Cover up (x – 1)^{2} with your fingertip and put in
x = Cover up (x – 2)^{2} and put x = So you get: Then : 2x + 1 = A(x – 1)(x – 2)^{2}
+ 3(x – 2)^{2} + B(x – 1)^{2}(x – 2) + 5(x – 1)^{2}. Equate x^{3}
coefficients, A
+ B = 0 Equate constant term, 1
= Solving, we get : A
= 8, B = 8. 
Example 4 The Keily’s Method is
also useful for repeated linear factors. This method is to use one factor at
a time, keeping the rest outside the expression. Point to note: Keily’s method
should be used with care if the fraction is improper during the process, as
in the following: Inside the bracket is
improper and division is used before applying the Keily’s Method. 