Exercise Zone : Pecahan Aljabar

Berikut ini adalah kumpulan soal mengenai Pecahan Aljabar tingkat dasar. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Terima kasih.

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No. 1

Tentukan nilai A, B dan C yang memenuhi identitas berikut.
{\dfrac{x^2+2x-2}{x^3+1}\equiv\dfrac{A}{x+1}+\dfrac{Bx+C}{x^2-x+1}}

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$\begin{array}{rl}{\displaystyle \frac{x^2+2x-2}{x^3+1}}& \equiv {\displaystyle \frac{A}{x+1}}+{\displaystyle \frac{Bx+C}{x^2-x+1}}\\ {\displaystyle \frac{x^2+2x-2}{x^3+1}}& \equiv {\displaystyle \frac{A(x^2-x+1)+(Bx+C)(x+1)}{(x+1)(x^2-x+1)}}\\ {\displaystyle \frac{x^2+2x-2}{x^3+1}}& \equiv {\displaystyle \frac{A{x}^2-Ax+A+B{x}^2+Bx+Cx+C}{x^3+1}}\\ {\displaystyle \frac{x^2+2x-2}{x^3+1}}& \equiv {\displaystyle \frac{(A+B)x^2+(-A+B+C)x+A+C}{x^3+1}}\end{array}$

A+B=1

$\begin{array}{rl}-A+B+C& =2\\ A-B-C& =-2\end{array}$

A+C=-2

$\begin{array}{rl}A+B& =1\\ A-B-C& =-2\\ A+C& =-2& +\\ 3A& =-3\\ A& =-1\end{array}$

$\begin{array}{rl}A+B& =1\\ -1+B& =1\\ B& =2\end{array}$

$\begin{array}{rl}A+C& =-2\\ -1+C& =-2\\ C& =-1\end{array}$

Jadi, A=-1, B=2, dan C=-1.