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

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


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