# Tag Info

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### Is there an easier way to prove the Fundamental Theorem of Algebra for polynomials with real coefficients?

The fundamental theorem of algebra is exactly as difficult for real vs. complex coefficients. The reason is that if $f(x) = f_0 + \dots + f_n x^n$ is any polynomial with complex coefficients, then the ...
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### Is there an easier way to prove the Fundamental Theorem of Algebra for polynomials with real coefficients?

The fundamental theorem of algebra is, in my opinion, the most striking example of a fact which is comprehensible and relevant very early on mathematically but which really has no simple explanation. ...
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### Does Newton's method converge for all polynomials?

As the other answers already noted, there are cases where the Newton iteration does not converge. One interesting question is for how "many" values this can occur and whether it's a set or ...
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### Is there an easier way to prove the Fundamental Theorem of Algebra for polynomials with real coefficients?

Though the following comments don't directly answer your stated question, perhaps they will be of some help. A field $F$ is said to be algebraically closed if all the roots of a polynomial with ...
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### Can't understand proof of fundamental theorem of symmetric polynomials

Let's take a look at a concrete example, say $g(x, y) = x + y + xy$. This is a symmetric polynomial, in fact it's the sum of a homogeneous symmetric polynomial $x + y$ of degree $1$ and another ...
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### May a trinomial with degree exceeding four have only real roots?

Yes, there are trinomials of arbitrarily large degree whose roots are all real, e.g., $$x^{n + 4} - (r^2 + s^2) x^{n + 2} + r^2 s^2 x^n = (x + s) (x + r) x^n (x - r) (x - s) , \quad r, s \in \Bbb R .$$...
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### Derivative as a matrix: $\mathbf{D}=\dfrac{\mathrm{d}}{\mathrm{d}x}$

Broadly speaking, yes. The derivative is a linear operator, meaning that $\frac{d}{dx}\left(a f(x) + b g(x)\right) = a \frac{d}{dx} f(x) + b \frac{d}{dx} g(x)$, as long as you are working in a vector ...
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### Finding the minimum of $f(x)=3x^2+4y^2+4xy-11x-6y$ where $x$ and $y$ are reals.

By direct calculation, one has \begin{align*} f(x,y) & = 3x^2+4y^2+4xy-11x-6y \\ & = (2y+x)^2 +2x^2 - 11x - 6y \\ & = \left(2y+x - \dfrac{3}{2} \right)^2 +2x^2-8x - \dfrac{9}{4} \\ & =...
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Let $\displaystyle f(a)=a^6-a^5+a^4-a^3+1$ $\bullet$ For $a\leq 0,$ Then $f(a)>0$ $\bullet$ For $0<a<1,$ Then $f(a)=a^6+a^4(1-a)+(1-a^3)>0$ $\bullet$ For $a\geq 1,$ Then $f(a)=a^5(a-1)+... • 5,122 10 votes Accepted ### prove that for any$|x|\leq 1$,$|f(x)|\leq 5/4$We can express the coefficients of$f(x)$in terms of$f(-1), f(0), f(1)$: $$f(x) = \left(\frac12 f(1) + \frac12 f(-1) - f(0)\right) x^2 + \left(\frac12f(1) - \frac12f(-1)\right) x + f(0).$$ (The ... • 37.8k 9 votes ### Prove that$x^{2020} + x^{1011} + 2 x^{1010} + x^2 + x + 1$does not have a real root Both$x^{1010}(x^{1010} + x + 2)$and$x^2 + x + 1$are non-negative functions of$x$, so that any real root of$x^{1010}(x^{1010} + x + 2) + x^2 + x + 1$must also be a real root of$x^2 + x + 1$. ... • 1,380 9 votes Accepted ### Show that, if$n$is even, then the polynomial$p(x) =x^n + x^{(n-1)}+...+x+1$does not have real roots. If$a$is a zero of$p(x)$, then it is a zero of$p(x)\cdot(x-1)=x^{n+1}-1$. It follows that$a^{n+1}=1$. If$a$was real, then necessarily$a=\pm 1$, though we can immediately exclude$a=-1$because$...
The question is about the irreducible polynomials of the ratio of theta-null values. The context is singular moduli. That is, define for convenience, $\,q_n:=\exp(-\pi\sqrt{n}).\,$ Then it is known ...