Find the value of the parameter $k$ (with $k0$) such that the rational function $\displaystyle \frac{x^{2}+4x+5}{(x+1)(x^{2}+k)}$ can be expressed in the form $\displaystyle \frac{A}{x+1}+\frac{Bx+C}{x^{2}+k}$ where $A,B,C$ are integers. Then give the complete partial‑fraction decomposition.

Question

Find the value of the parameter $k$ (with $k>0$) such that the rational function $\displaystyle \frac{x^{2}+4x+5}{(x+1)(x^{2}+k)}$ can be expressed in the form $\displaystyle \frac{A}{x+1}+\frac{Bx+C}{x^{2}+k}$ where $A,B,C$ are integers. Then give the complete partial‑fraction decomposition.

Accepted Answer

k = 1; $\displaystyle \frac{1}{x+1}+\frac{4}{x^{2}+1}$

Gce Model Questions Vol1 2019 Further Mathematics Question 26

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