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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
$$\dfrac{7t+4}{t^2+1} + \dfrac{t-3}{t^2+1} = \dfrac{7t+4+t-3}{t^2+1} = \dfrac{8t+1}{t^2+1}.$$
