Why partial fraction decomposition of $frac1s^2(s+2)$ is $fracAs+fracBs^2+fracC(s+2)$?Derivation of the general forms of partial fractionsWhy do you need two fractions for partial fraction decomposition with repeated factors?How can the correct form of the partial fractions decomposition be found for arbitrary rational functions?Integration - Partial Fraction DecompositionPartial Fraction Expansion of Transfer FunctionHow to solve Partial Fraction- Improper FractionsPartial Fraction Solution?Extra Square in Partial FractionLaurent Expansion partial fractionsComplicated partial fraction expansionIntegration of Partial Fraction ExpansionSimple partial fraction expansionConfusion with how partial fractions work

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Why partial fraction decomposition of $frac1s^2(s+2)$ is $fracAs+fracBs^2+fracC(s+2)$?


Derivation of the general forms of partial fractionsWhy do you need two fractions for partial fraction decomposition with repeated factors?How can the correct form of the partial fractions decomposition be found for arbitrary rational functions?Integration - Partial Fraction DecompositionPartial Fraction Expansion of Transfer FunctionHow to solve Partial Fraction- Improper FractionsPartial Fraction Solution?Extra Square in Partial FractionLaurent Expansion partial fractionsComplicated partial fraction expansionIntegration of Partial Fraction ExpansionSimple partial fraction expansionConfusion with how partial fractions work













1












$begingroup$


Can someone please explain why: $$frac1s^2(s+2)=fracAs+fracBs^2+fracC(s+2)$$



And not:$$frac1s^2(s+2)=fracAs^2+fracB(s+2)$$



I'm a bit confused where the extra s term comes from in the first equation.










share|cite|improve this question











$endgroup$











  • $begingroup$
    There are many answers available on MSE, i.e. here and here
    $endgroup$
    – callculus
    2 hours ago










  • $begingroup$
    More answers here too
    $endgroup$
    – David K
    35 mins ago















1












$begingroup$


Can someone please explain why: $$frac1s^2(s+2)=fracAs+fracBs^2+fracC(s+2)$$



And not:$$frac1s^2(s+2)=fracAs^2+fracB(s+2)$$



I'm a bit confused where the extra s term comes from in the first equation.










share|cite|improve this question











$endgroup$











  • $begingroup$
    There are many answers available on MSE, i.e. here and here
    $endgroup$
    – callculus
    2 hours ago










  • $begingroup$
    More answers here too
    $endgroup$
    – David K
    35 mins ago













1












1








1





$begingroup$


Can someone please explain why: $$frac1s^2(s+2)=fracAs+fracBs^2+fracC(s+2)$$



And not:$$frac1s^2(s+2)=fracAs^2+fracB(s+2)$$



I'm a bit confused where the extra s term comes from in the first equation.










share|cite|improve this question











$endgroup$




Can someone please explain why: $$frac1s^2(s+2)=fracAs+fracBs^2+fracC(s+2)$$



And not:$$frac1s^2(s+2)=fracAs^2+fracB(s+2)$$



I'm a bit confused where the extra s term comes from in the first equation.







algebra-precalculus partial-fractions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 mins ago









user21820

39.9k544159




39.9k544159










asked 2 hours ago









stuartstuart

1968




1968











  • $begingroup$
    There are many answers available on MSE, i.e. here and here
    $endgroup$
    – callculus
    2 hours ago










  • $begingroup$
    More answers here too
    $endgroup$
    – David K
    35 mins ago
















  • $begingroup$
    There are many answers available on MSE, i.e. here and here
    $endgroup$
    – callculus
    2 hours ago










  • $begingroup$
    More answers here too
    $endgroup$
    – David K
    35 mins ago















$begingroup$
There are many answers available on MSE, i.e. here and here
$endgroup$
– callculus
2 hours ago




$begingroup$
There are many answers available on MSE, i.e. here and here
$endgroup$
– callculus
2 hours ago












$begingroup$
More answers here too
$endgroup$
– David K
35 mins ago




$begingroup$
More answers here too
$endgroup$
– David K
35 mins ago










4 Answers
4






active

oldest

votes


















4












$begingroup$

The general result is the following.




Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
$$fracp(s)q(s)=fracr_1(s)q_1(s)+fracr_2(s)q_2(s) .$$




In your case the denominator factorises as $s^2$ times $s+2$ so you have
$$frac1s^2(s+2)=fracAs+Bs^2+fracCs+2 .$$
It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.






share|cite|improve this answer









$endgroup$




















    2












    $begingroup$

    If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $fracAas+b+fracB(as+b)^2+cdots+fracZ(as+b)^n$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $fracAs+fracBs^2$.






    share|cite|improve this answer









    $endgroup$








    • 1




      $begingroup$
      That's the rule, but I think the question was asking why is that the rule.
      $endgroup$
      – David K
      32 mins ago










    • $begingroup$
      @DavidK: Neither did your answer explain the rule you gave!
      $endgroup$
      – user21820
      1 min ago


















    2












    $begingroup$

    That is because for
    $$fracas^2+bs+cs^2(s+2)=fracAs^2+fracB(s+2),$$
    the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



    Or more simply, consider the example
    $$
    fracs+1s^2=frac1s^2+frac1s
    $$






    share|cite|improve this answer









    $endgroup$




















      1












      $begingroup$

      One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



      By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



      $$frac14(s+2)$$



      For large $s$ we can expand this in powers of $1/s$:



      $$frac14(s+2) = frac14 sfrac11+frac2s = frac14s - frac12 s^2 + mathcalOleft(frac1s^3right)$$



      The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



      $$frac12 s^2-frac14s $$



      The complete partial fraction expansion is thus given by:



      $$frac12 s^2-frac14s + frac14(s+2) $$






      share|cite|improve this answer









      $endgroup$













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        4 Answers
        4






        active

        oldest

        votes








        4 Answers
        4






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        4












        $begingroup$

        The general result is the following.




        Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
        $$fracp(s)q(s)=fracr_1(s)q_1(s)+fracr_2(s)q_2(s) .$$




        In your case the denominator factorises as $s^2$ times $s+2$ so you have
        $$frac1s^2(s+2)=fracAs+Bs^2+fracCs+2 .$$
        It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



        Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.






        share|cite|improve this answer









        $endgroup$

















          4












          $begingroup$

          The general result is the following.




          Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
          $$fracp(s)q(s)=fracr_1(s)q_1(s)+fracr_2(s)q_2(s) .$$




          In your case the denominator factorises as $s^2$ times $s+2$ so you have
          $$frac1s^2(s+2)=fracAs+Bs^2+fracCs+2 .$$
          It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



          Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.






          share|cite|improve this answer









          $endgroup$















            4












            4








            4





            $begingroup$

            The general result is the following.




            Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
            $$fracp(s)q(s)=fracr_1(s)q_1(s)+fracr_2(s)q_2(s) .$$




            In your case the denominator factorises as $s^2$ times $s+2$ so you have
            $$frac1s^2(s+2)=fracAs+Bs^2+fracCs+2 .$$
            It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



            Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.






            share|cite|improve this answer









            $endgroup$



            The general result is the following.




            Suppose that the degree of $p(s)$ is less than the degree of $q(s)$, and that $q(s)=q_1(s)q_2(s)$ where $q_1(s)$ and $q_2(s)$ have no common factor. Then there exist polynomials $r_1(s)$ and $r_2(s)$, with degrees less than $q_1(s)$ and $q_2(s)$ respectively, such that
            $$fracp(s)q(s)=fracr_1(s)q_1(s)+fracr_2(s)q_2(s) .$$




            In your case the denominator factorises as $s^2$ times $s+2$ so you have
            $$frac1s^2(s+2)=fracAs+Bs^2+fracCs+2 .$$
            It is then usually more convenient (though not obligatory) to split up the first fraction, which gives your answer.



            Note that you cannot, for the purposes of the above result, regard the denominator as $s$ times $s(s+2)$, because these polynomials do have a common factor.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 2 hours ago









            DavidDavid

            69.7k668131




            69.7k668131





















                2












                $begingroup$

                If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $fracAas+b+fracB(as+b)^2+cdots+fracZ(as+b)^n$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $fracAs+fracBs^2$.






                share|cite|improve this answer









                $endgroup$








                • 1




                  $begingroup$
                  That's the rule, but I think the question was asking why is that the rule.
                  $endgroup$
                  – David K
                  32 mins ago










                • $begingroup$
                  @DavidK: Neither did your answer explain the rule you gave!
                  $endgroup$
                  – user21820
                  1 min ago















                2












                $begingroup$

                If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $fracAas+b+fracB(as+b)^2+cdots+fracZ(as+b)^n$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $fracAs+fracBs^2$.






                share|cite|improve this answer









                $endgroup$








                • 1




                  $begingroup$
                  That's the rule, but I think the question was asking why is that the rule.
                  $endgroup$
                  – David K
                  32 mins ago










                • $begingroup$
                  @DavidK: Neither did your answer explain the rule you gave!
                  $endgroup$
                  – user21820
                  1 min ago













                2












                2








                2





                $begingroup$

                If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $fracAas+b+fracB(as+b)^2+cdots+fracZ(as+b)^n$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $fracAs+fracBs^2$.






                share|cite|improve this answer









                $endgroup$



                If your denominator has a factor of the form $(as+b)^n$ then to write partial fractions you should write all the powers up to $n$, i.e. $fracAas+b+fracB(as+b)^2+cdots+fracZ(as+b)^n$. In the case you showed, you have that $s^2$ is a factor of the denominator and that's why in partial fractions you should write the terms $fracAs+fracBs^2$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 2 hours ago









                Julian MejiaJulian Mejia

                39328




                39328







                • 1




                  $begingroup$
                  That's the rule, but I think the question was asking why is that the rule.
                  $endgroup$
                  – David K
                  32 mins ago










                • $begingroup$
                  @DavidK: Neither did your answer explain the rule you gave!
                  $endgroup$
                  – user21820
                  1 min ago












                • 1




                  $begingroup$
                  That's the rule, but I think the question was asking why is that the rule.
                  $endgroup$
                  – David K
                  32 mins ago










                • $begingroup$
                  @DavidK: Neither did your answer explain the rule you gave!
                  $endgroup$
                  – user21820
                  1 min ago







                1




                1




                $begingroup$
                That's the rule, but I think the question was asking why is that the rule.
                $endgroup$
                – David K
                32 mins ago




                $begingroup$
                That's the rule, but I think the question was asking why is that the rule.
                $endgroup$
                – David K
                32 mins ago












                $begingroup$
                @DavidK: Neither did your answer explain the rule you gave!
                $endgroup$
                – user21820
                1 min ago




                $begingroup$
                @DavidK: Neither did your answer explain the rule you gave!
                $endgroup$
                – user21820
                1 min ago











                2












                $begingroup$

                That is because for
                $$fracas^2+bs+cs^2(s+2)=fracAs^2+fracB(s+2),$$
                the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



                Or more simply, consider the example
                $$
                fracs+1s^2=frac1s^2+frac1s
                $$






                share|cite|improve this answer









                $endgroup$

















                  2












                  $begingroup$

                  That is because for
                  $$fracas^2+bs+cs^2(s+2)=fracAs^2+fracB(s+2),$$
                  the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



                  Or more simply, consider the example
                  $$
                  fracs+1s^2=frac1s^2+frac1s
                  $$






                  share|cite|improve this answer









                  $endgroup$















                    2












                    2








                    2





                    $begingroup$

                    That is because for
                    $$fracas^2+bs+cs^2(s+2)=fracAs^2+fracB(s+2),$$
                    the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



                    Or more simply, consider the example
                    $$
                    fracs+1s^2=frac1s^2+frac1s
                    $$






                    share|cite|improve this answer









                    $endgroup$



                    That is because for
                    $$fracas^2+bs+cs^2(s+2)=fracAs^2+fracB(s+2),$$
                    the left hand side has three parameters $a,b,c$, but the right hand side only has two parameters $a,b$. And if you try to solve TWO values from THREE equations, it will usually lead to a contradiction. So a third term of the right is needed. Even though this is not obvious in your question, you should think 1 as a degree 2 polynomial.



                    Or more simply, consider the example
                    $$
                    fracs+1s^2=frac1s^2+frac1s
                    $$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 2 hours ago









                    Holding ArthurHolding Arthur

                    1,360417




                    1,360417





















                        1












                        $begingroup$

                        One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



                        By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



                        $$frac14(s+2)$$



                        For large $s$ we can expand this in powers of $1/s$:



                        $$frac14(s+2) = frac14 sfrac11+frac2s = frac14s - frac12 s^2 + mathcalOleft(frac1s^3right)$$



                        The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



                        $$frac12 s^2-frac14s $$



                        The complete partial fraction expansion is thus given by:



                        $$frac12 s^2-frac14s + frac14(s+2) $$






                        share|cite|improve this answer









                        $endgroup$

















                          1












                          $begingroup$

                          One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



                          By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



                          $$frac14(s+2)$$



                          For large $s$ we can expand this in powers of $1/s$:



                          $$frac14(s+2) = frac14 sfrac11+frac2s = frac14s - frac12 s^2 + mathcalOleft(frac1s^3right)$$



                          The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



                          $$frac12 s^2-frac14s $$



                          The complete partial fraction expansion is thus given by:



                          $$frac12 s^2-frac14s + frac14(s+2) $$






                          share|cite|improve this answer









                          $endgroup$















                            1












                            1








                            1





                            $begingroup$

                            One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



                            By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



                            $$frac14(s+2)$$



                            For large $s$ we can expand this in powers of $1/s$:



                            $$frac14(s+2) = frac14 sfrac11+frac2s = frac14s - frac12 s^2 + mathcalOleft(frac1s^3right)$$



                            The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



                            $$frac12 s^2-frac14s $$



                            The complete partial fraction expansion is thus given by:



                            $$frac12 s^2-frac14s + frac14(s+2) $$






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                            $endgroup$



                            One can immediately see why in this case the partial fraction expansion will lead to a nonzero coefficient for the $1/s$ term. The asymptotic behavior of the fraction for large $s$ is $sim 1/s^3$. The singularity at $s = -2$ contributes a term proportional to $1/(s+2)$ to the partial fraction expansion, which for large $s$ behaves like $sim 1/s$. This $sim 1/s$ must be canceled out by the partial fraction expansion terms coming from the singularity at $s = 0$, this requires the presence of a contribution proportional to $1/s$.



                            By making this reasoning more precise we can get to the complete partial fraction expansion using only the contribution from the singularity at $s = -2$. The amplitude of the $1/(s+2)$ term in the partial fraction expansion is given by the factor that multiplies it in the fraction evaluated at $s = -2$, this is therefore equal to $1/4$. So the contribution to the partial fraction expansion coming from the singularity at $s = -2$ is:



                            $$frac14(s+2)$$



                            For large $s$ we can expand this in powers of $1/s$:



                            $$frac14(s+2) = frac14 sfrac11+frac2s = frac14s - frac12 s^2 + mathcalOleft(frac1s^3right)$$



                            The singularity at $s = 0$ will contribute terms to the partial fraction expansion whose large $s$ behavior will have to cancel out these first two terms, this means that this contribution to the partial fraction expansion is:



                            $$frac12 s^2-frac14s $$



                            The complete partial fraction expansion is thus given by:



                            $$frac12 s^2-frac14s + frac14(s+2) $$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










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