<s>[INST] Counting the Number of Real Roots of $y^{3}-3y+1$ Here's my question:

How many real roots does the cubic equation $y^3-3y +1$ have?

I graphed the function and it crossed the x-axis $3$ times. But my professor doesn't want a graphical explanation. So in that case, I was looking at the Fundamental Theorem of Algebra and states that a polynomial of degree  n  can have at most  n  distinct real roots. so therefore, there must be 3 real roots? 
EDIT
It seems that there are numerous ways to approach this problem after all. And we can expand this to other types of polynomials as well, not just cubics.
 [/INST] Set $x=2\cos\varphi$; then the equation becomes
$$
4\cos^3\varphi-3\cos\varphi=-\frac{1}{2}
$$
that is,
$$
\cos3\varphi=\cos\frac{2\pi}{3}
$$
Thus we get
$$
3\varphi=\frac{2\pi}{3}
\qquad\text{or}\qquad
3\varphi=\frac{2\pi}{3}+2\pi
\qquad\text{or}\qquad
3\varphi=\frac{2\pi}{3}+4\pi
$$
and therefore
$$
x=2\cos\frac{2\pi}{9}
\qquad\text{or}\qquad
x=2\cos\frac{8\pi}{9}
\qquad\text{or}\qquad
x=2\cos\frac{14\pi}{9}
$$
Note that choosing $3\varphi=-\frac{2\pi}{3}$ wouldn't give different solutions.
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