Resolve $\dfrac{x^2+x+1}{(x-1)^2(x+2)}$ into partial fractions.

Resolve (x²+x+1)/((x-1)²(x+2)) into partial fractions.

Exercise 7(a) · Q4

\dfrac{x^2+x+1}{(x-1)^2(x+2)}

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Step 1. Write

\frac{x^2+x+1}{(x-1)^2(x+2)}=\frac A{x-1}+\frac B{(x-1)^2}+\frac C{x+2}.

Step 2. Clear denominators:

x^2+x+1=A(x-1)(x+2)+B(x+2)+C(x-1)^2.

Step 3. Substitute $x=1$ : $1+1+1=B(3)\ \Rightarrow\ 3=3B\ \Rightarrow\ B=1$ .

Step 4. Substitute $x=-2$ : $4-2+1=C(9)\ \Rightarrow\ 3=9C\ \Rightarrow\ C=\dfrac13$ . …