Questions tagged [discriminant]

Discriminant of a polynomial $\;P\left(x\right) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{n}x^{n} \neq 0\,$ is defined as \begin{align} \Delta &= a_{n}^{2n-2}\prod_{ i < j } \big( r_i - r_j \big)^{2} = \left(-1\right)^{n\left(n-1\right)/2} a_{n}^{2n-2}\prod_{ i \neq j } \big( r_i - r_j \big) \end{align} where $\,r_1,\dots,r_n\,$ are roots of $P\left(x\right)$ (counting multiplicity)

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Relation of quadratic forms with discriminant $-p$ (prime) and $-4p$

I noticed a relationship between the size of class group of binary quadratic forms with discriminant $D=-p$ and $D=-4p$ with prime $p$: $$ \begin{aligned} p\equiv3\ (\text{mod }8) &\quad\...
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Given that $x_1, x_2, x_3$ are the roots of the polynomial $x^3-2x^2+3x+5=0$ find $(x_2-x_1)^2(x_3-x_1)^2(x_3-x_2)^2$.

Consider the polynomial: $$x^3-2x^2+3x+5=0$$ where $x_1, x_2$ and $x_3$ are the roots of the above polynomial. Now, consider the following determinant, which is defined using the above given roots: ...
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Why is $a_n(x) \neq 0$ for $a_n(x) = c_1 x a_{n-1}(x) + c_2 x a_{n-2}(x)$ if the discriminant of the characteristic polynomial $\Delta_{\lambda} > 0$?

Lets define the sequence $a_0 = 1$, $a_1 = c_1x$ and $a_n = c_1 x a_{n-1} + c_2 x a_{n-2}$ with $c_{1,2} \in \mathbb{N}$ and $x \in \mathbb{R}$. Then the characteristic polynomial is: $\lambda^2 - ...
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For the discriminant of a polynomial, how to prove $\Delta(f \bmod p) = \Delta(f) \bmod p$?

Let $f \in \mathbf{Z}[X]$ be a monic polynomial. I am trying to prove that $\Delta(f \bmod p) = \Delta(f) \bmod p$, where $\Delta$ is the discriminant of $f$, defined by $$ \Delta(f) = \prod_{1 \leq ...
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Inequality with x,y,z fractions $\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$

If $x,y,z>0$, show: $$\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$$ I expand and to prove $$x^3 - 2 x^2 y + x^2 z + x y^2 - x y z + y z^2\ge 0$$ I don't know how to do this.
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What is the condition that a cubic equation $x^3+ax^2+bx+c=0$ has exactly three positive real root?

What is the condition that a cubic equation $x^3+ax^2+bx+c=0$ has exactly three positive real root? If $G^2+4H^3<0$ then it has three real roots (where $G=c-ab/3+2a^2/27$, $H=b-a^2/9$). Then what ...
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Connection between an algebraic invariant and a maximisation issue

I came to be interested by the following rational function : $$f(x)=\dfrac{(x^2-x+1)^3}{x^2(x-1)^2}\tag{1}$$ while writing this answer ; I discovered that $f$ is connected to rather deep features of ...
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How to compute the value of the discriminant of $\mathcal{O}_K$

I found a $\mathbb{Z}$-basis $\left\{1, \sqrt{2}, \sqrt{-1}, (\sqrt{2}+\sqrt{-2})/2 \right\}$ for ring of integers $\mathcal{O}_K$ of some number filed $K$ and I need to compute the discriminant of $\...
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Algebraic and Trigonometric expression is $>0$ for all real $x$

Prove that $$2x^2\sin x+2x\cos x+2x^2+1$$ is always positive for all real $x$. From completing the square method Write $1$ as $\sin^2 x+\cos^2 x$ $$x^4+2x^2\sin x+\sin^2 x+x^2+2x\cos x+\cos^2 x+x^2-...
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A nontrvial inequality $\sum\limits_{\rm cyc}a^2 c(4a-3b-c)^2\ge 20abc\sum\limits_{\rm cyc}a(a-b)$

For real numbers $a,b,c>0$, show that $$\sum_{\rm cyc}a^2 c(4a-3b-c)^2\ge 20abc(a^2+b^2+c^2-ab-bc-ca)$$ This has the following dumbass notation: $$\begin{array}{ccccccccccc} &&&&...
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Trouble finding conditions for a function

I am having trouble solving this problem where I need to give conditions on $b$ and $c$ for the map $f \colon \mathbb{R} \rightarrow \mathbb{R}, f(x) = x^2 + bx + c$ to have a fixed point. Use these ...
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Discriminant of an quaternion algebra

Given an quaternion algebra $\mathcal{A} = (\alpha, \beta)_\mathbb{K}$ it is known that the $$\mathcal{D}(\mathcal{A}) = \prod \mbox{ideal primes where $\mathcal{A}$ is ramified}.$$ Ok, but I really ...
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Strange outcomes for ring of integers in Sage

The number field is $K=\mathbb{Q}(\alpha)=\mathbb{Q}[x]/(f)$ with $\alpha=\sqrt[4]{24}$ and $f=x^4-24$. Now Sage tells me that $\Delta(f)=-2^{17}\cdot3^3$ and $\Delta_K:=\Delta(\mathscr{O}_K)=-2^{11}\...
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Discriminant of $X=Z(f)$ in terms of join, intersection and projection.

Say we have $X\subset \mathbb{CP}^{n}$ a projective variety, we may even assume it is a hypersurface for convenience. Let $\pi:\mathbb{CP}^{n}\to\mathbb{CP}^{n-1}$ be the standard projection. I am ...
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Roots of quadratic eqaution lies in an AP

I tried finding the common difference between the roots but didnt know what to do next. if someone could tell me how to solve this or give a headstart itll be a lot helpful
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Finding discriminant of a monic polynomial.

I have now engaged in studying Galois Theory from NPTEL online lecture series which encompasses Finite Fields and Galois Theory. While watching the $48$-th lecture on Discriminant of a Polynomial a ...
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Writing Cubic Equation in terms of discriminant (with possible shifts and translations)

So I noticed this fact for the following fact for quadratic equations. I need one notation that if one equation can be gotten from another through a shift or scaling of variable then I will denote ...
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Find the ring of integers of $\mathbb{Q}(\theta)$

I was trying to find the ring of integers of $\mathbb{Q}(\theta)$, where $\theta^3 -2\theta + 2 = 0$. I compute the discriminant of the basis $\{1, \theta, \theta^2\}$, but unfortunately it is $-4*19$,...
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Can we find the discriminant of an Equation of any degree?

Some days ago I was solving a question which gave me a hard time. After doing some research I found out that it required the Discriminant of a Cubic Equation. I looked up to the internet and I found ...
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$\mathbb{Q}(\sqrt{n})$ is Contained in the Cyclotomic Field of the $4n$'th Primitive Root of Unity.

From Silverman and Tate, Rational Points on Elliptic Curves. Exercise 6.1 Let $\zeta'$ be the $4n$'th primitive root of unity. Use (c) to prove that $\mathbb{Q}(\sqrt{n})$ is contained in the ...
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Cube root of complex numbers and treatment of cubic equations

Let $Z^3=a+bi$, $b\not=0$ Is it possible for such a complex number to have real roots for $Z=\sqrt[3]{a+bi}$? I did some calculations and found out that it is not but I have a problem: The equation $...
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What is discriminant and how is it derived for cubic equations?

I have been studying on cubic equations and I have actually reached the cubic formula on my own, but I couldn't really understand what a discriminant is. I obtained the discriminant of a quadratic ...
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How to find discriminant of a cubic equation?

I am studying on cubic equations for an essay and I have reached the general formula for any cubic equation. However I didn't realise what is what while formulating it, like discriminant. Now, I am ...
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discriminants in ring of integers

I've been trying to solve the following problem: Find the ring of integer of $\mathbb{Q}(\theta)$ when $\theta^3 + \theta + 1 = 0$. I started by computing the discriminant, which is $-31$. Then I ...
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Finding integral basis of $K=\Bbb Q(\theta)$

Let $m$ be a cubefree integer. Set $m=hk^2$, where $h$ is square free, so that $k$ is square-free and $(h,k)=1$. Set $\theta=m^{1/3}$ and $K=\Bbb Q(\theta)$. Then an integral basis for $K$ is $$\{1,\...
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Find all the possible integer values of $a$ such that the equation about $x$: $(a+1)x^2-(a^2+1)x+2a^3-6=0$ has all integer roots

Find all the possible integer values of $a$ such that the equation about $x$: $(a+1)x^2-(a^2+1)x+2a^3-6=0$ has all integer roots. I've been given this as a homework problem and haven't been able to ...
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Find range of values of k for which f(x) <= k for all real values of x

Find the range of values of k for which f(x) <=k for all real values of x f(x) = -2x^2 + 8x + 17 -2x^2 + 8x + 17 <= k 2x^2 - 8x -17 + k >=0 I ...
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Find the range of values of $k$ for which $kx^2 + 8x + k <6$ for all real values of $k$

Find the range of values of $k$ for which $kx^2 + 8x + k <6 $ for all real values of $k$. I'm unsure if the discriminant must be greater than zero or less than zero. My working steps: \begin{...
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Condition on the discriminant of the second derivative of a biquadratic function to be concave upwards

Question: For which values of 'a' will the function $f(x)=x^4+ax^3+\frac3 2 x^2+1$ will be concave upward along the entire real line. My Approach: I know that the concavity (concave upwards or ...
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Solve: $x$, $y$ are real, and $(x-3)^2+(y-3)^2=6$, find the maximum value of $\frac{y+1}{x+2}$

Solve: $x$, $y$ are real, and $(x-3)^2+(y-3)^2=6$, find the maximum value of $\frac{y+1}{x+2}$ This question is from a class I'm taking, and the current topic is discriminants. I found the ...
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Solve: The equation $7x^2-(R+13)x+R^2-R-2=0$ has one real root in range (0,1) and one real root in range (1,2). What's the range of R? [closed]

Solve: The equation $7x^2-(R+13)x+R^2-R-2=0$ has one real root in range (0,1) and one real root in range (1,2). What's the range of R? I've been given this problem for a math class (the topic was ...
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square root in cyclotomic field and discriminant

I am trying to solve the following exercice: Let $ω = e^{2πi/ p}$ with $p$ an odd prime. Show that $\Bbb Q[ω]$ contains $\sqrt p$ if $p ≡ 1(mod 4)$, and $\sqrt {− p}$ if $p ≡ −1 (mod 4)$. ...
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Information about the discriminant given a quadratic extension of fields

If we have a quadratic number field $K=\mathbb{Q}(\sqrt{d})$ then its discriminant is $$\Delta_K = \begin{cases} d & d\equiv 1 \pmod 4\\ 4d & d \equiv 2,3 \pmod 4 \end{cases}.$$ In particular,...
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Discriminant Analysis: the polynomial kernel of Support Vector Machine

The following 【Quiz】 and 【Official Answer】are the rough translation (with minor modification) of Quiz No.03-1 of the exam of the "2018's semi-first grade of Japan Statistical Society Certificate (JSSC)...
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$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}-3\geq k\left ( \frac{x^{2}+y^{2}+z^{2}}{xy+yz+zx}-1 \right )$

Let $x,y,z>0$. Find the maximum value of $k$ such that the inequality $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}-3\geq k\left ( \frac{x^{2}+y^{2}+z^{2}}{xy+yz+zx}-1 \right )$ is true for all $x,y,z>0$....
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Given $a$, $b$ are integers with $a > b$ and the two roots $\alpha$, $\beta$ of the equation $3x^2 + 3(a+b)x + 4ab = 0$ satisfy a relation

Given $a$, $b$ are integers with $a > b$ and the two roots $\alpha$, $\beta$ of the equation $3x^2 + 3(a+b)x + 4ab = 0$ satisfy the relation $$\alpha(\alpha + 1) + \beta(\beta + 1) = (\alpha + 1)(\...
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$\frac{2n+3}{n^2+n+1}$ How many integers are there that make an expression an integer?

$$\frac{2n+3}{n^2+n+1}$$ How many integers are there that make an expression an integer? $-1,0,2$ It is seen that the numbers make the fraction integer immediately. But the answer key says there ...
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Discriminant of homogeneous polynomials

Let $f$ be a homogeneous polynomial in variables $x,y,z$. Suppose that the sum of coefficients of $\frac{\partial^i f}{\partial x^i}$ is $0$ for each $0 \leq i \leq r$. I believe that, in this ...
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Find all the integer solutions involving the Diophantine-equation $\frac{x+ y}{x^{2}+ y^{2}+ xy}= \frac{5}{19}$ .

Find all the integer solutions involving the following Diophantine-equation $$\frac{x+ y}{x^{2}+ y^{2}+ xy}= \frac{5}{19}$$ By W|A I found that $2$ and $3$ are the solutions of the O-p, my observation ...
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High School Exam Question: Straight Line Intersecting with Absolute Value Line(s), and Discriminants

Here's a question I've encountered in a recent high school examination. Find the range of values of m such that the line $y=mx-3$ intersects with the graph of $ y=2-|3x - 5|$ at exactly two points....
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Given numbers $a,b,c\geqq0$ and $-\frac{2}{11}\leqq k\leqq0$. Prove that $(k+1)^{6}(a+b+c)^{2}(\!ab+bc+ca\!)^{2}-81\prod\limits_{sym}(ka+b)\geqq0$ .

Problem. Given three numbers $a, b, c\geqq 0$ and $k= constant$ so that $- \dfrac{2}{11}\leqq k\leqq 0$. Prove that : $$(\!k+ 1\!)^{6}(\!a+ b+ c\!)^{2}(\!ab+ bc+ ca\!)^{2}\!- 81(ka+ b)(kb+ a)(kb+ c)(...
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How to interpret “has a solution” in this exercise?

Is the following question poorly written? I suppose it is a translation since it comes from a Japanese institution. I am referring to the assumption "... has a solution". Given this information ...
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205 views

Under condition $2x^2 + y^2= 4$ for real numbers $x, y$, find the maximum and minimum value of $4x + y^2$.

How can I solve this problem. I can find absolute maximum and minimum value of equation with given interval. But here, I don't know where should I start. Could you explain step by step?!
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3answers
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Discriminants and quadratics algebra problem

Let $D_1$ be the discriminant of $a(x+2)^2+b(x+2)+c=0$. How does it compare to $D_2$, the discriminant of $ax^2+bx+c=0$?
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Problem in a question concerning Geometric progression and quadratic equations

Consider the following problem:- If $a,b,c,d,p$ are distinct real numbers such that $(a^2+b^2+c^2)p^2 -2p(ab+bc+cd)+(b^2+c^2+d^2)≤0$ then prove that $a,b,c,d$ are in geometric progression. The ...
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Prove $x+ y+ z= 3,\,x^{\,2}+ y^{\,2}+ z^{\,2}= 9\,\therefore\,y- x\leqq 2\sqrt{3}$ .

Prove $$x+ y+ z= 3,\,x^{\,2}+ y^{\,2}+ z^{\,2}= 9\,\therefore\,y- x\leqq 2\sqrt{3}$$ I have a solution, and I'm looking forward to seeing a nicer one(s), thanks for your interests a lot ! We have $$(\,...
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show that $\mathbb{Q}(\sqrt[n]{2}) \neq \mathbb{Q}(\sqrt[n]{3})$

I would like to show that $\mathbb{Q}(\sqrt[n]{2}) \neq \mathbb{Q}(\sqrt[n]{3})$ for an even $n$. I was hinted that I should use the following fact (which I already proved): If $L/\mathbb{Q}$ finite ...
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0answers
55 views

How to classify the factorization of polynomials over finite fields?

I want to study the factorization of a specific polynomial with coefficients over finite fields. Let $f\in \mathbb{Z}$ a polynomial, I define the polynomial class as follows: suppose $(\bar{f})\in \...
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1answer
52 views

How to factor a polynomial over finite fields?

We want to study the factorization of a specific polynomial with coefficients over finite fields. Let $f\in \mathbb{Z}$ a polynomial, we define the polynomial class as follows: suppose $(\bar{f})$ $\...
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9answers
180 views

Minimize this real function on $\mathbb{R}^{2}$ without calculus?

When it comes to minimizing a differentiable real function, calculus comes into play immediately. If $f: (x,y) \mapsto (x+y-1)^{2} + (x+2y-3)^{2} + (x+3y-6)^{2}$ on $\mathbb{R}^{2}$, and if one is ...

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