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A bit of a beginner question, but I've been told that between 2 x-intercepts, for any polynomial of degree 2 or higher. That is true. But the controversy here is that apparently, it has to be exactly in the middle between the 2. I'm not talking about quadratics; the only turning point is at $x = -b/2a$. I am talking about polynomials of higher degree. For example, $y= x^4−2x^2+x$. It seems that I am wrong, even for most polynomials: the turning point is usually not in the middle.

1. Is this completely false? (That it has to be in the middle) Or is it the case for specific types of graphs?

2. If it is, then what is the standard rule for finding the turning point between any 2 x-intercepts for any given polynomial


I am just in year 10, so please bear with me if this question is too simple.

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It's not true in general. Consider $y=x^3-x$, which has turning points at $x=\pm\sqrt{3}/3$.

As far as I know, there isn't a nice condition for when it will be true.

Find the turning points of a polynomial function $y=f(x)$ amounts to solving $f'(x)=0$, where $f'$ is the derivative of $f$. $f'$ is a polynomial with degree one less than the degree of $f$. Solving polynomials of high degree is difficult in general.

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  1. It is completely false: the turning point can be anywhere between the x-intercepts. There can also be more than one turning point.

  2. The standard way for finding the turning points is to find the zeros of the derivative of the polynomial.

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