What are the rules to make partial fractions?

Question

I've got four in my book but sometimes i don't get the answer by simply following the rule this happens especially with substitution method of Partial fractions of A and B.

Genesis Allen · Accepted Answer

Be careful, it can be a little complicated I'll go through a few examples since there are countless problems with their own solution. Say we have #(f(x))/(g(x)^n)# We need to write it as a sum. #(f(x))/(g(x)^n)=sum_(a=1)^nA/(g(x)^a)# For example, #(f(x))/(g(x)^3)=A/(g(x))+B/(g(x)^2)+C/(g(x)^3)# Or, we have #(f(x))/(g(x)^ah(x)^b)=sum_(n_1=1)^aA/(g(x)^(n_1))+ sum_(n_2=1)^bB/(h(x)^(n_2))# For example, #(f(x))/(g(x)^2h(x)^3)=A/(g(x))+B/(g(x)^2)+C/(h(x))+D/(h(x)^2)+E/(h(x)^3)# The next bit can't be written as a generalised formula, but you have to follow simple fraction addition to combine all the fractions into one. Then you multiply both sides by the denominator which leaves you with #f(x)="A summation of A, B, C, ... along with functions"# Now, you have to use values of #x# which leaves one letter from #"A, B, C, D, ..."# on its own and rearrange to find its value, continue to find other letters until you have to perform simultaneuous equations, etc. For example:
#(f(x))/(g(x)h(x)^2)=A/(g(x))+B/(h(x))+C/(h(x)^2)# #(f(x))/(g(x)h(x)^2)=A/(g(x))+(Bh(x)+C)/(h(x)^2)# #(f(x))/(g(x)h(x)^2)=(Ah(x)^2+g(x)(Bh(x)+C))/(h(x)^2)# #f(x)=Ah(x)^2+Bh(x)g(x)+Cg(x)# Now, find a value for #x# such that #h(x)=0#, let's call this #a# #f(a)=Ah(a)^2+Bh(a)g(a)+Cg(a)# #f(a)=Cg(a)# #C=(f(a))/(g(a))# Now, find a value for #x# such that #g(x)=0#, let's call this #b#. Also, put in your value for #C#. #f(b)=Ah(b)^2+Bh(b)g(b)+(f(a))/(g(a))g(b)# #f(b)=Ah(b)^2# #A=(f(b))/(h(b)^2)# #f(x)=(f(b))/(h(b)^2)h(x)^2+Bh(x)g(x)+(f(a))/(g(a))g(x)# Just use any value for #x# such that #x!=a and x!=b#, let's call this #c# #f(c)=(f(b))/(h(b)^2)h(c)^2+Bh(c)g(c)+(f(a))/(g(a))g(c)# #Bh(c)g(c)=f(c)-(f(b))/(h(b)^2)h(c)^2+(f(a))/(g(a))g(c)# #B=(f(c)-(f(b))/(h(b)^2)h(c)^2+(f(a))/(g(a))g(c))/(h(c)g(c))# Put your values for #A, B and C# into:
#(f(x))/(g(x)h(x)^2)=A/(g(x))+B/(h(x))+C/(h(x)^2)#

Hazel Anglin · Answer

undefined To decompose a rational function into partial fractions, follow these steps:

1. Factor the denominator completely.
2. Write the rational function as a sum of fractions, one for each distinct factor of the denominator.
3. For each factor of the denominator, set up an equation where the numerator of the partial fraction is a polynomial of one degree less than the factor's degree.
4. Solve the equations to find the numerators of the partial fractions.
5. Combine the partial fractions into a single expression.