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

Resolve (x² - x + 1)/((x + 1)(x - 1)²) into partia – Class 12 Mathematics

Question 8 of 8

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

Yanam BieapBIEAP Intermediate Board 2022Subjective· 4mImportance★★★★★

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$\dfrac{x^2-x+1}{(x+1)(x-1)^2}=\dfrac{3}{4(x+1)}+\dfrac{1}{4(x-1)}+\dfrac{1}{2(x-1)^2}.$

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

Multiply through: $x^2-x+1=A(x-1)^2+B(x+1)(x-1)+C(x+1).$

Put $x=1$ : $\ 1=C(2)\Rightarrow C=\tfrac12.$

Put $x=-1$ : $\ 3=A(4)\Rightarrow A=\tfrac34.$ …