What are the rules to make partial fractions?

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.

1 Answer
Feb 22, 2018

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)#