# Tagged Questions

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### hexic polynomial question

I am faced with a polynomial of the form $$ax^6+bx^3+cx+d=0,$$ where the coefficients are complex. I want to be able to say something about the roots of this polynomial (including finding them!). Is ...
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### Uniform convergence of a sequence of polynomial logarithm

Let $P\in \Bbb{C}[X]$ of degree $d\ge 2$. For $n\in \Bbb{N}$ (include $O$). Denote by $P^n$ the $n$-th composition and $g_n: z\mapsto \frac{1}{d^n}\log(\max \{1,\vert P^n\vert\})$. Show that ...
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### A “repeated roots allowed” version of the continuity of roots

Let $R_n$ denote the set of all monic real polynomials of degree $n$ all of whose roots are real. Then $R_n$ is a closed subset of the $n+1$-dimensional space ${\mathbb R}_n[X]$. For $P\in R_n$, ...
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### Kantorovich Theorem example

I need to write in C a program that finds roots of a 6th order polynomial. I was thinking of using Kantorovich Theorem convergence of Newton's method to find when can I use Newton-Rhapson method. I'm ...
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### Real polynomials, complex zeroes and the Intermediate value theorem

I have a second grad polynomial p(x). For arguments sake lets say $$p(x) = x^2 + 16x + 76$$ I also have an inequation $$p(x) > 0$$ Now the inequation does not have a real solution, but only ...
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### Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
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### What is a rational way of factoring polynomials?

Consider the following polynomial $$P_x:=x^4+1$$ I want to represent it as a product of two polynomials with real coefficients of grade $2$. I have done this using, call it, a brute force method. ...
While trying to solve this question my testing lead to an observation that I found interesting in its own right. Consider the linear transformation $L:P\to P$ from the space of polynomial functions ...