I need to calculate the number of real solutions in a polynomial equation. From the Fundamental Theorem of Algebra, I know that an equation of degree n has exactly n roots (but, here the complex ones are also included). And I just need to determine how many real ones.

For example, I am asked to the following equation:


From the Fundamental Algebra Theorem, it is known to have 5 roots (real and complex). Next, I have made a sketch of the function and it comes out that it only cuts the X-axis once between zero and one. Then, this equation would have a real solution and 4 complex solutions. But is there any way to solve it analytically without having to graph the function?

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    $\begingroup$ Hint: will have at least one real root because of odd degree of the polynomial. Then consider the derivative $15(x^4+1) >0$, so the function is increasing. $\endgroup$ – Anurag A Dec 25 '18 at 8:58
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    $\begingroup$ According to en.wikipedia.org/wiki/Descartes%27_rule_of_signs there's exactly one and that zero is positive. $\endgroup$ – Michael Hoppe Dec 25 '18 at 19:00

Let $P(x)=3x^5+15x-8$. Since $\lim_{x\to\infty}P(x)=\infty$ and $\lim_{x\to-\infty}P(x)=-\infty$, it follows from the intermediate value theorem that $P(x)$ has at least one real root. Now, imagine that it has two distinct roots $x_1$ and $x_2$. Then, by Rolle's theorem, there is some $y$ between $x_1$ and $x_2$ such that $P'(y)=0$. But $P'(y)=15y^4+15$, and therefore it can't be $0$.


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