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Quiz 17
1.) For which values of $x$ is the following expression defined? $$\dfrac{3x^2+2x+1}{5x+2}.$$ Solution: It is defined for all $x$ with the property that $$5x+2 \neq 0,$$ i.e. $x \neq -\dfrac{2}{5}$.

2.) For which values of $x$ is the following expression defined? $$\dfrac{2x+1}{x^2+5x+6}.$$ Solution: It is defined for all $x$ with the property that $$x^2+5x+6 \neq 0,$$ i.e. $$(x+3)(x+2) \neq 0,$$ i.e. $x+3 \neq 0$ or $x+2 \neq 0$. Therefore $x \neq -3$ and $x \neq -2$.

3.) Multiply and simplify $$\dfrac{(t+5)(2t+6)}{5t+3} \cdot \dfrac{10t+6}{(t-5)(2t+6)}.$$ Solution: Multiply to get $$\begin{array}{ll} \dfrac{(t+5)(2t+6)}{5t+3} \cdot \dfrac{10t+6}{(t-5)(2t+6)} &= \dfrac{2(t+5)(t+3)}{5t+3} \cdot \dfrac{2(5t+3)}{2(t-5)(t+3)} \\ &=\dfrac{2(t+5)}{(t-5)} \end{array}$$

4.) Multiply and simplify $$\left( \dfrac{t^2-25}{t^2+8t+7} \right) \left( \dfrac{t+7}{t-5} \right).$$ Solution: Multiply to get $$\begin{array}{ll} \left( \dfrac{t^2-25}{t^2+8t+7} \right) \left( \dfrac{t+7}{t-5} \right) &= \left( \dfrac{(t-5)(t+5)}{(t+7)(t+1)} \right) \left( \dfrac{t+7}{t-5} \right) \\ &=\dfrac{t+5}{t+1}. \end{array}$$

5.) Add $$\dfrac{7t+4}{t^2+1} + \dfrac{t-3}{t^2+1}.$$ Solution: Compute $$\dfrac{7t+4}{t^2+1} + \dfrac{t-3}{t^2+1} = \dfrac{7t+4+t-3}{t^2+1} = \dfrac{8t+1}{t^2+1}.$$