Cubic's Real Roots

A cubic equation has real roots if the y-value of the first extremum is the same sign as the leading coefficient and the y-value of the second extremum is the opposite sign as the leading coefficient.

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The extrema: $\frac{dy}{dx}=9x^2-25=0$ , $\displaystyle x=-\frac53$ is the local maximum (first extremum) and $\displaystyle x=\frac53$ is the local minimum (second extremum)

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• $\displaystyle3\left(-\frac53\right)^3-25\left(-\frac53\right)+n>0$ and $\displaystyle3\left(\frac53\right)^3-25\left(\frac53\right)+n>0$
• $\displaystyle n>-27\frac79$ and $\displaystyle n<27\frac79$

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Therefore there are $\fbox{55}$ possible integers $n$ .

Applying Sturm's theorem to get the number of real roots of a polynomial, it is needed to compose a sequence, where the first polynomial comes from the statement, the second through the first derivative of the first polynomial and, with recurrence, the n-th polynomial is the remainder of the polynomial long division of a term by its sucessive term, multiplied by -1. These are the terms of Sturm sequence for the polynomial on the statement:

$\left \{ p_0(x)=3x^{3}-25x+n\\ p_1(x)=9x^{2}-25\\ p_2(x)=\frac{50}{3}x-n\\ p_3(x)=25-\frac{81}{2500}n^{2} \right \}$

Evaluating each polynomial to the positive and negative infinities:

$\left \{ \text{Negative infinity evaluating} \\ \lim_{x \rightarrow -\infty} {p_0(x)=x^{3}(3-\frac{25}{x^{2}}+\frac{n}{x^{3}})= - \infty}\\ \lim_{x \rightarrow -\infty} {p_1(x)=+ \infty}\\ \lim_{x \rightarrow -\infty} {p_2(x)=- \infty}\\ \lim_{x \rightarrow -\infty} {p_3(x) \rightarrow \text{depends on n}} \right \}$

$\left \{ \text{Positive infinity evaluating} \\ \lim_{x \rightarrow +\infty} {p_0(x)=+ \infty}\\ \lim_{x \rightarrow +\infty} {p_1(x)=+ \infty}\\ \lim_{x \rightarrow +\infty} {p_2(x)=+ \infty}\\ \lim_{x \rightarrow +\infty} {p_3(x) \rightarrow \text{depends on n}} \right \}$

For the case in which the fourth function on the Sturm sequence is positive, there'll be three changes of signal on evaluating the negative infinity, and none on evaluating the positive infinity. Thus, by Sturm's theorem, it must be three real roots for the fourth function as a positive value. Otherwise, if we set the fourth polynomial as negative, the number of changes of signal would imply in only one real root. When discussing the number of real roots by Sturm's theorem, the multiplicities are already discounted.

$p_3(x)>0 \rightarrow 25>\frac{81}{2500}n^{2} \rightarrow n^{2}< \frac{5^{4}\cdot 10^{2}}{3^{2}}$ $\pm n< \frac{250}{3} \rightarrow -\frac{250}{3}<n<\frac{250}{3}$

Since n must be an integer number, it lies on the interval below, which has the following number of terms:

$S= \# \{ n \in \mathbb{Z} \mid n \in [-27, 27] \}=27-(-27)+1=\boxed {\text{55 integer possible values of n}}$