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UIC / Mathematics / MATH 181 / What is the first step in any partial fraction problem?

What is the first step in any partial fraction problem?

What is the first step in any partial fraction problem?

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.lst-kix_92tou2eydnyc-0 > li:before{content:"● "}.lst-kix_92tou2eydnyc-1 > li:before{content:"○ "}ol.lst-kix_ykrv90m20537-5.start{counter-reset:lst-ctn-kix_ykrv90m20537-5 0}.lst-kix_mr6wn5a51d5v-2 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-2}.lst-kix_ykrv90m20537-6 > li{counter-increment:lst-ctn-kix_ykrv90m20537-6}.lst-kix_92tou2eydnyc-6 > li:before{content:"● "}ol.lst-kix_1f6lfdtukceq-0.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-0 0}.lst-kix_mr6wn5a51d5v-1 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-1}.lst-kix_92tou2eydnyc-4 > li:before{content:"○ "}.lst-kix_92tou2eydnyc-5 > li:before{content:"■ "}.lst-kix_92tou2eydnyc-2 > li:before{content:"■ "}.lst-kix_92tou2eydnyc-3 > li:before{content:"● "}.lst-kix_ykrv90m20537-7 > li{counter-increment:lst-ctn-kix_ykrv90m20537-7}ol.lst-kix_1f6lfdtukceq-3.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-3 0}ol.lst-kix_ykrv90m20537-2.start{counter-reset:lst-ctn-kix_ykrv90m20537-2 0}.lst-kix_mr6wn5a51d5v-3 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-3}.lst-kix_1f6lfdtukceq-0 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-0}.lst-kix_mr6wn5a51d5v-0 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-0}ol.lst-kix_mr6wn5a51d5v-5.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-5 0}.lst-kix_ykrv90m20537-8 > li{counter-increment:lst-ctn-kix_ykrv90m20537-8}.lst-kix_ykrv90m20537-5 > li{counter-increment:lst-ctn-kix_ykrv90m20537-5}ol.lst-kix_mr6wn5a51d5v-2.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-2 0}ol.lst-kix_ykrv90m20537-0.start{counter-reset:lst-ctn-kix_ykrv90m20537-0 0}.lst-kix_ykrv90m20537-3 > li{counter-increment:lst-ctn-kix_ykrv90m20537-3}ol.lst-kix_mr6wn5a51d5v-4.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-4 0}.lst-kix_1f6lfdtukceq-7 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-7}.lst-kix_1f6lfdtukceq-1 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-1}.lst-kix_mr6wn5a51d5v-5 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-5}ol.lst-kix_1f6lfdtukceq-6.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-6 0}ol.lst-kix_ykrv90m20537-7.start{counter-reset:lst-ctn-kix_ykrv90m20537-7 0}ol.lst-kix_ykrv90m20537-1.start{counter-reset:lst-ctn-kix_ykrv90m20537-1 0}ol.lst-kix_mr6wn5a51d5v-6{list-style-type:none}ol.lst-kix_ykrv90m20537-8.start{counter-reset:lst-ctn-kix_ykrv90m20537-8 0}ol.lst-kix_mr6wn5a51d5v-7{list-style-type:none}ol.lst-kix_mr6wn5a51d5v-8{list-style-type:none}ol.lst-kix_mr6wn5a51d5v-2{list-style-type:none}ol.lst-kix_mr6wn5a51d5v-3{list-style-type:none}ol.lst-kix_1f6lfdtukceq-5.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-5 0}ol.lst-kix_mr6wn5a51d5v-4{list-style-type:none}ol.lst-kix_mr6wn5a51d5v-5{list-style-type:none}ol.lst-kix_mr6wn5a51d5v-0{list-style-type:none}ol.lst-kix_mr6wn5a51d5v-1{list-style-type:none}.lst-kix_ykrv90m20537-0 > li{counter-increment:lst-ctn-kix_ykrv90m20537-0}.lst-kix_1f6lfdtukceq-4 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-4}.lst-kix_mr6wn5a51d5v-7 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-7}.lst-kix_ykrv90m20537-1 > li{counter-increment:lst-ctn-kix_ykrv90m20537-1}.lst-kix_mr6wn5a51d5v-8 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-8}ol.lst-kix_mr6wn5a51d5v-6.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-6 0}.lst-kix_1f6lfdtukceq-1 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-1,lower-roman) ". "}ol.lst-kix_mr6wn5a51d5v-3.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-3 0}.lst-kix_1f6lfdtukceq-0 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-0,lower-latin) ". "}ol.lst-kix_ykrv90m20537-6.start{counter-reset:lst-ctn-kix_ykrv90m20537-6 0}.lst-kix_1f6lfdtukceq-3 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-3}ol.lst-kix_1f6lfdtukceq-1.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-1 0}.lst-kix_1f6lfdtukceq-6 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-6}ol.lst-kix_ykrv90m20537-3.start{counter-reset:lst-ctn-kix_ykrv90m20537-3 0}ol.lst-kix_ykrv90m20537-8{list-style-type:none}ol.lst-kix_ykrv90m20537-6{list-style-type:none}ol.lst-kix_ykrv90m20537-7{list-style-type:none}ol.lst-kix_ykrv90m20537-4{list-style-type:none}ol.lst-kix_1f6lfdtukceq-4.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-4 0}ol.lst-kix_ykrv90m20537-5{list-style-type:none}ol.lst-kix_ykrv90m20537-2{list-style-type:none}.lst-kix_ykrv90m20537-2 > li{counter-increment:lst-ctn-kix_ykrv90m20537-2}ol.lst-kix_1f6lfdtukceq-7.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-7 0}ol.lst-kix_ykrv90m20537-3{list-style-type:none}ol.lst-kix_ykrv90m20537-0{list-style-type:none}ol.lst-kix_ykrv90m20537-1{list-style-type:none}.lst-kix_mr6wn5a51d5v-6 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-6}.lst-kix_mr6wn5a51d5v-2 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-2,lower-roman) ". "}.lst-kix_mr6wn5a51d5v-1 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-1,lower-latin) ". "}.lst-kix_mr6wn5a51d5v-3 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-3,decimal) ". "}ul.lst-kix_92tou2eydnyc-8{list-style-type:none}.lst-kix_mr6wn5a51d5v-0 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-0,decimal) ". "}.lst-kix_mr6wn5a51d5v-4 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-4,lower-latin) ". "}ol.lst-kix_1f6lfdtukceq-2.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-2 0}ol.lst-kix_mr6wn5a51d5v-1.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-1 0}ol.lst-kix_1f6lfdtukceq-0{list-style-type:none}.lst-kix_ykrv90m20537-6 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-6,decimal) ". "}.lst-kix_ykrv90m20537-4 > li{counter-increment:lst-ctn-kix_ykrv90m20537-4}.lst-kix_mr6wn5a51d5v-8 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-8,lower-roman) ". "}.lst-kix_mr6wn5a51d5v-4 > li{counter-increment:lst-ctn-kix_mr6wn5a51d5v-4}.lst-kix_ykrv90m20537-7 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-7,lower-latin) ". "}ol.lst-kix_mr6wn5a51d5v-8.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-8 0}ol.lst-kix_1f6lfdtukceq-5{list-style-type:none}.lst-kix_mr6wn5a51d5v-6 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-6,decimal) ". "}ol.lst-kix_1f6lfdtukceq-6{list-style-type:none}ol.lst-kix_1f6lfdtukceq-7{list-style-type:none}.lst-kix_mr6wn5a51d5v-5 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-5,lower-roman) ". "}.lst-kix_mr6wn5a51d5v-7 > li:before{content:"" counter(lst-ctn-kix_mr6wn5a51d5v-7,lower-latin) ". "}ol.lst-kix_1f6lfdtukceq-8{list-style-type:none}.lst-kix_ykrv90m20537-8 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-8,lower-roman) ". "}ol.lst-kix_1f6lfdtukceq-1{list-style-type:none}ol.lst-kix_1f6lfdtukceq-2{list-style-type:none}ol.lst-kix_1f6lfdtukceq-3{list-style-type:none}ol.lst-kix_1f6lfdtukceq-4{list-style-type:none}ol.lst-kix_1f6lfdtukceq-8.start{counter-reset:lst-ctn-kix_1f6lfdtukceq-8 0}.lst-kix_1f6lfdtukceq-2 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-2,decimal) ". "}ol.lst-kix_mr6wn5a51d5v-0.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-0 0}.lst-kix_1f6lfdtukceq-3 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-3,lower-latin) ". "}.lst-kix_1f6lfdtukceq-4 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-4,lower-roman) ". "}ol.lst-kix_ykrv90m20537-4.start{counter-reset:lst-ctn-kix_ykrv90m20537-4 0}.lst-kix_ykrv90m20537-5 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-5,lower-roman) ". "}.lst-kix_1f6lfdtukceq-5 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-5}.lst-kix_1f6lfdtukceq-6 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-6,lower-latin) ". "}.lst-kix_1f6lfdtukceq-2 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-2}.lst-kix_92tou2eydnyc-7 > li:before{content:"○ "}.lst-kix_1f6lfdtukceq-5 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-5,decimal) ". "}.lst-kix_ykrv90m20537-4 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-4,lower-latin) ". "}.lst-kix_92tou2eydnyc-8 > li:before{content:"■ "}.lst-kix_1f6lfdtukceq-8 > li{counter-increment:lst-ctn-kix_1f6lfdtukceq-8}.lst-kix_ykrv90m20537-3 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-3,decimal) ". "}ul.lst-kix_92tou2eydnyc-4{list-style-type:none}ul.lst-kix_92tou2eydnyc-5{list-style-type:none}ul.lst-kix_92tou2eydnyc-6{list-style-type:none}.lst-kix_ykrv90m20537-0 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-0,decimal) ". "}.lst-kix_ykrv90m20537-2 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-2,lower-roman) ". "}ul.lst-kix_92tou2eydnyc-7{list-style-type:none}ul.lst-kix_92tou2eydnyc-0{list-style-type:none}.lst-kix_1f6lfdtukceq-7 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-7,lower-roman) ". "}.lst-kix_1f6lfdtukceq-8 > li:before{content:"" counter(lst-ctn-kix_1f6lfdtukceq-8,decimal) ". "}ul.lst-kix_92tou2eydnyc-1{list-style-type:none}ul.lst-kix_92tou2eydnyc-2{list-style-type:none}ol.lst-kix_mr6wn5a51d5v-7.start{counter-reset:lst-ctn-kix_mr6wn5a51d5v-7 0}.lst-kix_ykrv90m20537-1 > li:before{content:"" counter(lst-ctn-kix_ykrv90m20537-1,lower-latin) ". "}ul.lst-kix_92tou2eydnyc-3{list-style-type:none}

In general f, g are polynomials  

  • Degree of f degree of g

ie  must do polynomial divisionDon't forget about the age old question of What are the layers of the Earth?

If you want to learn more check out What happens to price and quantity of golf club membership as wealth increases?

  1. Distinct Factors

  1. Repeated factors

Don't forget about the age old question of What are the regulatory mechanisms?

  1. Irreducible Factor

If you want to learn more check out What does range measure?

Don't forget about the age old question of What is a political ideology?

7.5 Partial Fractions

Question 1. True or False?

        If you want to learn more check out Discuss the two major periods for the cell.

                        

Question 2. The first step in any partial fraction problem is to factor the denominator. Consider the rational expression                        (x-4)(x+2)

                                                

                        

If we factor the denominator, then we find that

        

Question 3. Which method will be most efficient in solving the integral

                

Substitution                                                =

In general f, g are polynomials

Degree of f degree of g                

        ie.  must do a polynomial division

Question4. How can we integrate

        

Question 5. Setup the partial fraction decomposition for:

  1.                                 

        

When x = -2        -2=        0        +        0        +-        8C = C =¼

When x = 6        6=        64A        +        0        +        0 = A = 3/32

When x = 0        4A        4a        -        12B        -6C        = B =

To solve:

  1. multiply time common denom.
  2. Plug in #s for x, good choices would com from common denom

Question 6. Bonus question!  True or false?

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