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Evaluate $\dfrac{x^{3}+1}{x^{2}-x+1}$ if $x=7,85$ without using a calculator. Show your work.
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Evaluate $\dfrac{x^{3} + 1}{x^{2} - x + 1}$ if $x = \text{7,85} \text{ without using a calculator. Show your work.}$

First simplify the expression:

\begin{align*} \frac{x^{3} + 1}{x^{2} - x + 1} & = \frac{(x + 1)(x^2 - x + 1)}{x^2 - x + 1} \\ & = x + 1 \end{align*}

Now substitute the value of $x$: $\text{7,85} + 1 = \text{8,85}$.

With what expression must $(a - 2b)$ be multiplied to get a product of $(a^{3} - 8b^{3})$?

$(a - 2b)(a^2 + 2ab + 4b^2) = a^3 - 8b^3$

So, the expression is $a^2 + 2ab + 4b^2$.

With what expression must $27x^{3} + 1$ be divided to get a quotient of $3x + 1$?

\begin{align*} 27x^3 + 1 & = (3x + 1)(9x^2 - 3x + 1)\\ \frac{(3x + 1)(9x^2 - 3x + 1)}{9x^2 - 3x + 1} &= 3x + 1 \end{align*}

Therefore the expression is $9x^2 - 3x + 1$.

What are the restrictions on the following?

$\dfrac{4}{3x^2 + 2x - 1}$
\begin{align*} \frac{4}{3x^2 + 2x - 1} &= \frac{4}{(3x - 1)(x+1)} \\ x \ne \frac{1}{3} &\text{ and } x \ne -1 \end{align*}
$\dfrac{a}{3(b-a) + ab - a^2}$
\begin{align*} \frac{a}{3(b-a) + ab - a^2} &= \frac{a}{3(b-a) + a(b-a)} \\ &=\frac{a}{(b-a)(a+3)} \\ a \ne b &\text { and } a \ne -3 \end{align*}
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