### 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 finger-tip and put in x = 1 in what is left: this gives 3.

Cover up (x – 2)2 and put x = 2 in what is left: this gives 5.

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 x3 coefficients,         A + B = 0

Equate constant term,          1 = -4A - 2B + 17

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.