55 votes
Accepted

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 ...
Qiaochu Yuan's user avatar
52 votes

Does Newton's method converge for all polynomials?

Here is a plot of the function $$f(x) = 49 x^7+31 x^6-10 x^5-41 x^4+37 x^3-21 x^2-9 x+12$$ in the region of interest. The blue curve is $f$. The orange lines represent the forward orbit of $x_0 = \...
heropup's user avatar
  • 136k
51 votes
Accepted

Is the notion "If a polynomial has small coefficients (relative to the exponent), then it has small roots" true?

There exist estimates for the size of the largest root. The most general go back to the idea that $z$ is not a root of $$ p(z)=a_nz^n+a_{n-1}+...+a_1z+a_0 $$ if $|z|>R>0$ with an outer root ...
Lutz Lehmann's user avatar
32 votes

Is the notion "If a polynomial has small coefficients (relative to the exponent), then it has small roots" true?

No. Take a polynomial with all roots "very large" (according to the context), and scale it down by dividing by an even larger number. To employ the counterexample in the comments, take $p(x) ...
D S's user avatar
  • 5,015
25 votes

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. ...
Noah Schweber's user avatar
21 votes

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 ...
emacs drives me nuts's user avatar
20 votes

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 ...
Timothy Chow's user avatar
19 votes

Is $\cos$ a polynomial function of $\sin$?

There is no polynomial (or any function) $P$ such that $\cos(x) = P(\sin(x))$ for all $x \in \Bbb R$. Because that function would have to satisfy $$ P(0) = P(\sin(0)) = \cos(0) = 1 $$ and $$ P(0) = ...
Martin R's user avatar
  • 113k
19 votes
Accepted

How did Artin discover the function $f(x)=\frac{(x^2-x+1)^3}{x^2(x-1)^2}$ with the properties $f(x)=f(1-x)=f(\frac{1}{x})$?

Note that $g(x) = 1-x $ and $h(x) = \frac{1}{1-x}$ generate a subgroup $G$ of the Möbius group $\mathrm{Aut}(\hat{\mathbb{C}})$ that is isomorphic to $S_3$ via the relations: $$ g^{2} = \mathrm{id}, \...
Sangchul Lee's user avatar
17 votes
Accepted

The rational function $\tan\left(n\arctan x\right)$

Consider the complex number $z = 1 + ix$ $$z = |z|e^{i\theta} = |z|(\cos \theta + i\sin \theta)$$ $$\frac {z}{|z|\cos\theta} = (1+i\tan\theta) = z$$ $$\tan \theta = x$$ $$\theta = \arctan x$$ $z^n = |...
user317176's user avatar
  • 11.2k
17 votes

Strange behaviour of $x^2+5x+7$ under iteration

We will never achieve eventual divisibility with $x^2+5x+7$. Cases $p=2$ and $p=3$ fail easily, so we must accept $p>3$ and then reckon with the cases below. Suppose $p=3k+1$. Then we will have an &...
Oscar Lanzi's user avatar
  • 39.5k
16 votes
Accepted

Does there exist a field where all even degree equations have solutions but not all odd degree equations?

No. Suppose that every polynomial over $F$ of even degree has a root in $F$, and pick $p \in F[x]$ of odd degree. Then, $\deg (p^2)$ is even, so $p(\alpha)^2 = 0$ for some $\alpha \in F$. But that ...
Travis Willse's user avatar
15 votes
Accepted

How can we solve the particular equation $16x^5-200x^3-200x^2+25x+30=0$ in closed form?

This polynomial $f$ has Galois group $F_5$, the Frobenius group of order 20. Since this group is solvable, then $f$ is solvable by radicals. Here are Magma commands showing this. ...
Viktor Vaughn's user avatar
15 votes

Is it possible to prove that the polynomial $x^4 - 2x^3 + 16$ doesn't have a real zero by writing it as a sum of squares?

$$ x^4-2x^3+16=\frac{1}{2}x^2(x-2)^2+\frac{1}{2}(x^2-2)^2+14 $$
Gonçalo's user avatar
  • 9,334
15 votes
Accepted

Is $\cos$ a polynomial function of $\sin$?

This polynomial $P$ would have to satisfy $$ z^2 + P(z)^2 = 1 $$ But if $P$ had degree $d > 1$, $z^2 + P(z)^2$ would have degree $2d$. And degree $0$ or $1$ clearly doesn't work.
Robert Israel's user avatar
13 votes

Generalizing Radicals to Solve Quintics and Above

For degree five, $\{\,x^2,x^3,x^5,x^5+x\,\}$ is such a set. Given $a$, a solution of $x^5+x=a$ is called a Bring radical, and if you allow the operation of taking Bring radicals, you can solve any ...
Gerry Myerson's user avatar
12 votes
Accepted

Is $X^4-3X^2+2X+1$ irreducible over the rationals?

Alternative method. Over $\mathbb{Z}_3$ we see $x=-1$ is a root and we find the factorization into irreducibles $f=(x^3+2x^2+x+1)(x+1)$. Now if $f$ factors over $\mathbb{Z}$, it too must factor as ...
Sil's user avatar
  • 16.6k
12 votes
Accepted

Why does it seem like that a tangent line of an odd-degree polynomial function crosses the curve at more than one point?

I think your misunderstanding is that "a tangent line only crosses a curve at one specific point". If a line $L$ is tangent to a curve $C$ at a point $P$, then $L$ will meet $C$ at $P$ and ...
David's user avatar
  • 82.7k
11 votes
Accepted

How to prove that $f(x,y)=xy(x+y)$ isn't surjective as $f:\mathbb{Q}^2\to\mathbb{Q}$.

With elementary reasoning we can reduce to Fermat's Last Theorem for $n=3$, which in turn does have an elementary proof (see e.g. Euler's proof). That is, first write $x=\frac{r_1}{s_1}$, $y=\frac{r_2}...
M W's user avatar
  • 9,866
11 votes
Accepted

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 ...
Qiaochu Yuan's user avatar
11 votes
Accepted

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., $$x^{n + 4} - (r^2 + s^2) x^{n + 2} + r^2 s^2 x^n = (x + s) (x + r) x^n (x - r) (x - s) , \quad r, s \in \Bbb R .$$...
Travis Willse's user avatar
10 votes

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 ...
ConMan's user avatar
  • 24.3k
10 votes

Find the max/min value of $x^2+y^2$ subject to $3x^2+5xy+3y^2=1$

The problem may be simpler in polar coordinates. Our constraint is: $$3x^2+5xy+3y^2=1$$ $$3(x^2+y^2)+5xy=1$$ $$3r^2+5(r \cos \theta)(r \sin \theta)=1$$ $$3r^2+5r^2\cos\theta\sin\theta=1$$ $$3r^2+\...
Dan's user avatar
  • 15.2k
10 votes

A tricky system of non-linear multivariate equations

Add the first equation to twice the second and to the third and express the result as $$ (x+a)^2+(y+b)^2=(z+c)^2 $$ Draw the following triangles to illustrate this equation as well as the equations $$...
John Wayland Bales's user avatar
10 votes

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} \\ & =...
TheSilverDoe's user avatar
  • 29.7k
10 votes

Prove that $a^6-a^5+a^4-a^3+1>0$

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)+...
jacky's user avatar
  • 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 ...
River Li's user avatar
  • 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 $x^2 + x + 1$. ...
terran's user avatar
  • 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 $...
Zuy's user avatar
  • 4,666
9 votes

Ratio of theta functions as roots of polynomials

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 ...
Somos's user avatar
  • 35.3k

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