Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x

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1 Setio Notes Pge Prtil Frtio Deompositio Suppose we were sked to write the followig s sigle frtio: We would eed to get ommo deomitors: You will get: Distributig o top will give you: 8 This simplifies to: Therefore we ome to the followig: I this setio we will be doig the reverse of wht we just did We will strt with the sigle frtio d brek it up deompose it ito seprte frtios This is somethig you will do i lulus ourse Sometimes it is esier to do lulus opertios o two smller frtios isted of oe big frtio Three Rules of How Frtio Deomposes Let P be polyomil Rule : P Rule : P Rule : b b b b P Where b is irreduible, or oftorble EXMPLE: Set up the followig for deompositio but DO NOT SOLVE: 6 We first eed to ftor: Now we will use rule : EXMPLE: Set up the followig for deompositio but DO NOT SOLVE: This time we t ftor the deomitor This mes we eed to use rule :

2 EXMPLE: Set up the followig for deompositio but DO NOT SOLVE: This oe requires rule : EXMPLE: Set up the followig for deompositio but DO NOT SOLVE: Sometimes you my eed to ombie more th oe rule This oe will use rule d rule : Setio Notes Pge EXMPLE: Determie the prtil frtio deompositio: The first thig we should do with ll these kid of problems is to ftor if possible: Now we 7 use rule to set it up: Now we wt to get ommo deomitors with the right side of the equtio: 7 This will give us: 7 Sie the deomitor of eh frtio is the sme, we just set the umertors equl to eh other You will get the followig equtio: 7 From here the book shows two methods of solvig this I will oly fous o oe method This oe ivolves hoosig vlue for d pluggig it ito both sides of the equtio You wt to hoose umber tht will el prt of the equtio so you oly hve oe vrible left to solve for I the problem bove I wt to let be d - Let We will put ito both sides of the equtio: 7 You wt to simplify iside the prethesis first: 0 6 This simplifies further to: 6 Solvig for 7 Let - We will put ito both sides of the equtio: 7 You wt to simplify iside the prethesis first: 6 0 This simplifies further to: 6 Solvig for we get: Our swer is writte s: 7 7

3 EXMPLE: Determie the prtil frtio deompositio: 6 Setio Notes Pge The first thig we should do is to ftor the deomitor This ivolves the groupig method Ftor the first two terms d the seod two terms seprtely: Now ftor out the ommo ftor of : We ftor this oe more time to get: So ow our problem 6 beomes: Now we use rule to set it up: 6 Now we wt to get ommo deomitors This will give us: 6 6 Sie the deomitor of eh frtio is the sme, we just set the umertors equl to eh other You will get the followig equtio: 6 You wt to hoose umber tht will el prt of the equtio so you oly hve oe vrible left to solve for I the problem bove I wt to let be,, d - Let We will put i for : 6 You wt to simplify iside the prethesis first: 0 0 This simplifies further to: Solvig for we get: / Let We will put i for : 6 You wt to simplify iside the prethesis first: 0 0 This simplifies further to: Solvig for we get: / Let - We will put i for : 6 You wt to simplify iside the prethesis first: 0 0 This simplifies further to: Solvig for we get: 7 / 6 Our swer is writte s: 6 / 7 / 6 / or EXMPLE: Determie the prtil frtio deompositio: 7 We use rule to set this up: Now we wt to get ommo deomitors with

4 7 the right side of the equtio: This will give us: Setio Notes Pge 7 Sie the deomitor of eh frtio is the sme, we just set the umertors equl to eh other You will get the followig equtio: 7 You wt to hoose umber tht will el prt of the equtio so you oly hve oe vrible left to solve for I the problem bove I wt to let be Let We will put ito both sides of the equtio: 7 You wt to simplify iside iside the prethesis first: 0 This simplifies further to: So ow our problem beomes: 7 We do t hve other vlue to plug i for to el somethig out So ow we let be NY umber It does t mtter whih oe you hoose beuse you will get the sme swer for Let s let 0 sie this is esy oe to plug i: Let 0 We will put 0 ito both sides of the equtio: 70 0 You wt to simplify iside the prethesis first: 0 Solvig for we get: Our swer is writte s: EXMPLE: Determie the prtil frtio deompositio: We wt to ftor ommo ftor out of the deomitor We will get: Now we use rule d rule to set it up: Now we wt to get ommo deomitors This will give us: Sie the deomitor of eh frtio is the sme, we just set the umertors equl to eh other You will get the followig equtio: You wt to hoose umber tht will el prt of the equtio so you oly hve oe vrible left to solve for I the problem bove I wt to let be 0, d Let 0 We will put 0 i for : You wt to simplify iside the prethesis first: 0 0 This simplifies further to: Solvig for we get:

5 Setio Notes Pge Let We will put i for : You wt to simplify iside the prethesis first: 0 0 This gives us: So ow our problem beomes: Sie we hve ru out of umbers to plug i for to el terms, we ow hoose NY umber to plug i I will hoose Let We will put i for : You wt to simplify iside the prethesis first: This simplifies further to: 0 Solvig for we get: Our swer is writte s: EXMPLE: Determie the prtil frtio deompositio: 6 9 We wt to ftor ommo ftor out of the deomitor We will get: We ftor this 6 9 oe more time: Now we use rule d rule to set it up: Now we wt to get ommo deomitors: This will give us: Settig the umertors equl we get: Now hoose vlues for Let 0 We will put 0 i for : This gives us 9, so / Let - We will put - i for : This gives us:, so So ow our problem beomes: / Sie we hve ru out of umbers to plug i for to el terms, we ow hoose NY umber to plug i I will hoose Let We will put i for : / This gives us 8 Solvig for we get / / / Our swer is:, or you write:

6 EXMPLE: Determie the prtil frtio deompositio: 7 Setio Notes Pge 6 7 We wt to ftor ommo ftor out of the deomitor We will get: Now we use rule 7 d rule to set it up: 7 This will give us: Now we wt to get ommo deomitors 7 Settig the umertors equl we get: Now hoose vlues for 7 Let 0 We will put 0 i for : This gives us 7, so 7 7 So ow our problem beomes: 7 Sie we hve ru out of umbers to plug i for to el terms, we ow hoose NY umber to plug i I will hoose 7 Let We will put i for : 7 This gives us We write this s: 6 This equtio be redued to: 8 ut we do t hve eough iformtio to solve for the two ukows, so we eed oe more equtio We eed to pik other vlue for I will let 7 Let We will put i for : 7 This gives us 6 6 We write this s: 6 60 This equtio be redued to 8 We ow eed to solve the system: If we subtrt the two equtios we get 7, so 7 / Goig to the top equtio we hve 7 / 8, or 7 8 The Our swer is: 7 7 / 7 /, or you write: 7 7 7

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