Back to the class 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
$$\begin{array}{ll}
\dfrac{(t+5)(2t+6)}{5t+3} \cdot \dfrac{10t+6}{(t-5)(2t+6)} &= \dfrac{2(t+5)(t+3)}{5t+3} \cdot \dfrac{2(5t+3)}{2(t-5)(t+3)} \\
&=\dfrac{2(t+5)}{(t-5)}
\end{array}$$