Question 1.33: Express as a single fraction x²+3x+2/x²-1 - 2/2x+6...
Express as a single fraction
\frac{x^2+3 x+2}{x^2-1}-\frac{2}{2 x+6}Step-by-Step
Each fraction is written in its simplest form:
\begin{aligned} \frac{x^2+3 x+2}{x^2-1} & =\frac{(x+1)(x+2)}{(x-1)(x+1)}=\frac{x+2}{x-1} \\ \frac{2}{2 x+6} & =\frac{2}{2(x+3)}=\frac{1}{x+3}\end{aligned}The l.c.d. is (x-1)(x+3). Each fraction is written in an equivalent form with l.c.d. as denominator:
\frac{x+2}{x-1}=\frac{(x+2)(x+3)}{(x-1)(x+3)}, \quad \frac{1}{x+3}=\frac{x-1}{(x-1)(x+3)}So
\begin{aligned} \frac{x^2+3 x+2}{x^2-1}-\frac{2}{2 x+6} & =\frac{x+2}{x-1}-\frac{1}{x+3} \\ & =\frac{(x+2)(x+3)}{(x-1)(x+3)}-\frac{(x-1)}{(x-1)(x+3)} \\& =\frac{(x+2)(x+3)-(x-1)}{(x-1)(x+3)} \\& =\frac{x^2+5 x+6-x+1}{(x-1)(x+3)} \\& =\frac{x^2+4 x+7}{(x-1)(x+3)}\end{aligned}Related Answered Questions
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