College Algebra, Chapter 1, 1.5, Section 1.5, Problem 18

Find all real solutions of the equation $7x^3 -x + 1 = x^3 + 3x^2 + x$


$
\begin{equation}
\begin{aligned}

7x^3 -x + 1 =& x^3 + 3x^2 + x
&& \text{Given}
\\
\\
7x^3 - x^3 - 3x^2 - x - x + 1 =& 0
&& \text{Combine like terms}
\\
\\
6x^3 - 3x^2 - 2x + 1 =& 0
&& \text{Group terms}
\\
\\
(6x^3 - 3x^2) - (2x - 1) =& 0
&& \text{Factor out } 3x^2
\\
\\
3x^2(2x - 1) - (2x - 1) =& 0
&& \text{Factor out } 3x^2 - 1
\\
\\
(3x^2 - 1)(2x - 1) =& 0
&& \text{Zero Product Property}
\\
\\
3x^2 - 1 =& 0 \text{ and } 2x - 1 = 0
&& \text{Solve for } x
\\
\\
x =& \pm \sqrt{\frac{1}{3}} \text{ and } x = \frac{1}{2}
&&

\end{aligned}
\end{equation}
$