I was practicing some maths from the UKMT website's mentoring scheme sheets and I found this question on the sample for sheet 9:
On the same axes, sketch the graphs
$y=(x+1)^3-(x+1)$
$y^2=(x+1)^3-(x+1)$
I also have to identify the points where the graphs intersect.
I found that $(x+1)^3-(x+1) = x(x+1)(x+2)$
From there I found three intersection points: $(0,0), (-1,0)$ and $(-2,0)$
I need to solve $x(x+1)(x+2) = 1$ for another intersection point, but I cant think of any ways to solve it. I was also had no idea on how to sketch a graph for $y^2 = (x+1)^3 - (x+1)$
Can it be solved easily? $$(x+1)^3-(x+1)-1=x^3+3x^2+2x-1$$ The cubic has no rational roots, so there is no simple expression for the $x$-coordinate of the fourth intersection (although some exact expression can be written down). The $x$-coordinate is about $0.324717957\dots$, but the $y$-coordinate is just $1$.