cover-up rule.pdf

The Cover-up Rule – or how to make partial fractions easy What is it? The cover-up rule is a technique for streamlining the calculations when setting up partial fractions. It is simpler to apply than the usual long-winded technique and as a result makes for more confident and so more accurate work. How does it work? Here is an example of the technique in use. My aim is to show just how simple a complex problem can be made. At this stage I make no attempt to explain or justify the technique. Just watch how it works. ( ) ( ) ( ) ? + + + + + ? +? +? + ? ? = + + + + + = + ? + + + ii 2 32 2613 9 32 3 4 11 3 32 2 8 13 6 2 2 1 3 2 62 632 26 6 ( 3)( 4) 2 2 1 3 2 1 2 2 2 2 1 3 2 x x x x x x x x x x x Here is the alternative by the standard procedure. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + = + + + + + + + + ? + + = + + + + + + + + 2 2 8 13 6 2 2 1 3 2 2 2 1 3 2 8 13 6 2 1 3 2 2 3 2 2 2 1 x x A B C x x x x x x x x A x x B x x C x x Now set successively = ?2x , = ? 1 2 x , = ? 2 3 x : ? = ? ? ? + = ? ? + + ? = ? = = ? ? ? + = + + ? = = ? ? ? + = + + ? = ? i i 13 31 1 2 2 2 2 32 262 4 1 3 9 3 3 3 2 32 26 6 ( 3)( 4) .0 .0 12 12 1 2 6 .0 .0 2 6 .0 .0 2 x A B C A A x A B C B x A B C C so that the partial fractions are + ? + + + 1 2 2 2 2 1 3 2x x x as before. That is all much longer, more writing and time-consuming. So how do you do it? The cover-up rule can be applied to calculate the coefficients for simple linear factors of the denominator. Here all three are simple. Here is the initial set of calculations again. ( ) ( ) ( ) + + + + + ? + ? ?= + + + = + ? + 28 13 6 2 2 1 3 2 32 26 6 ( 3)( 4) ... ... 2 1 ... ... 2 x x x x x x x Take the first one, x + 2. What is the value which makes that bracket (x + 2) zero? x = -2. Cover up the factor (x + 2) in the original expression and put x = -2 everywhere else. ( ) ( ) ( ) =? + + + + + 2 2 8 13 6 2 1 3 22 x x xx x x which evaluates to ? + = ? ? 32 26 6 1 ( 3)