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# Repeated Roots & Derivatives

we know, real roots are governed by the intersection of any real function with the $$x$$-axis. But, how do we know weather this intersected function with the x-axis at point $$x_{0}$$ has more than one root at that point.
$$f(x)= (x-3)^3 (x-2)^2 (x-1)x$$
which has the derivatives
$${f}'(x)= 108+1161x^2-648x+$$ $$400x^4-952x^3-84x^5+7x^6$$

$${f}''(x)=-648-2856x^2+2322x-$$ $$420x^4+1600x^3+42x^5$$

$${f}'''(x)=2322+4800x^2-5712x$$+ $$210x^4-1680x^3$$

And of course we knew intuitively that $$f(x)$$ has the following real roots $$3,3,3$$$$2,2$$, $$1,0$$ .We will study the repeated roots here which are 3 and 2.

Solving the derivatives with respect to $$x$$ will yield the following:

$${f}'(x)$$ has zeros when $$x$$ =  $$3$$ , $$3$$ , $$2$$ , $$2.454295852$$ , $$1.270906755$$ , $$0.2747973935$$.

$${f}''(x)$$ has zeros when $$x$$ = $$3$$ , $$2.698648167$$ , $$2.203645141$$ , $$1.533922047$$ , $$0.5637846449$$.

$${f}'''(x)$$ has zeros when $$x$$ =$$2.878374203$$ , $$2.445752631$$ , $$1.806349702$$ , $$0.8695234637$$.

In General:

Let $$x_{0} \in \mathbb{R}$$ be any arbitrary root for the polynomial function $$f(x)$$  i.e. $$f(x_{0}) = 0$$ then $$x_{0}$$ said to be repeated root $$n+1$$ times if $$\exists m,n \in \mathbb{N}$$  such that $$\forall m \leq n$$ $$f^{(m)}(x_{0}) = 0$$

Exercise: Let $$p$$ and $$q$$ be real numbers with $$p \leq 0$$. Find the co-ordinates of the turning points of the cubic $$y=x^3+px+q$$. Show that the cubic equation $$x^3+px+q$$ has three real roots, with two or more repeated, precisely when $$4p^3+27q^2=0$$
Under what conditions on $$p$$ and $$q$$  does $$x^3+px+q=0$$ have (i) three distinct real roots, (ii) just one real root? How many real roots does the equation $$x^3+px+q=0$$ have when $$p \geq 0$$ ?
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Does this Method imply trigonometric function and other function or restricted to polynomials ?

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