2
votes
2answers
72 views

Zeroes of polynomial

$$c_1,c_2 \text{ are polynomial's }g(x)=x^2+ax+b \text{ roots } \Leftrightarrow \begin{cases} g(c_1)=c_1^2+ac_1+b=0 \\ g(c_2)=c_2^2+ac_2+b=0 \end{cases}$$ Prove that for every polynomial with integer ...
2
votes
2answers
53 views

finding the value of u of equation 5u^2 = 10u

I was solving a question, and while solving that problem I noticed something $5u^2 = 10u$ (solving this) this can be solved as: $5 \cdot u \cdot u = 10 \cdot u$ $u = \dfrac{10u}{5u}$ $u = 2$ ...
1
vote
1answer
36 views

Basic question from LA: Why do we find Roots of a polynomial?

This may sound like a basic question, but I am sorry to say that I did not find it's answer which completely satisfy my query. Here is the question: "What is the need to find roots of a polynomial ?" ...
1
vote
2answers
40 views

Finding root for the segment - found the formula but it doesn't work for some values - wrong formula?

I have the segment, defined as $(x_1, y_1)$, $(x_2, y_2)$. I know that $y_1\ge 0$ and $y_2 < 0$. I want to compute the root point for that segment. I decided to do it that way: ...
3
votes
2answers
57 views

If $x\in\mathbb R$, solve $x^{\lfloor x\rfloor}=\frac{9}{2}$, where $\lfloor x\rfloor$ is the integer part of $x$.

If $x\in\mathbb R$, solve $x^{\lfloor x\rfloor}=\frac{9}{2}$, where $\lfloor x\rfloor$ is the integer part of $x$. Of course, $x=\lfloor x\rfloor+\{x\}$, where $\{x\}$ is the fractional part of ...
-2
votes
3answers
54 views

A question on quadratic equations.. Given below in the picture.

PLease also tell how u got to the answer as I want to know the way to solve further questions
3
votes
1answer
94 views

Finding real cubic root of the equation

The cubic equation has one real root.Find it. $\displaystyle 8x^3-3x^2-3x-1=0$
1
vote
4answers
148 views

Eigenvalues and the Characteristic Equation

Given the following matrix, $$ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} $$ assuming eigenvectors exist for $A$, they can be found by first solving for $\lambda$ ...
1
vote
2answers
251 views

what is the maximum number of roots of quadratic function with 3 variables?

Given the general quadratic form with $3$ variables $(x,y,z):ax^2 + by^2 +cz^2 + dxy + eyz +fzx + gx + hy + iz = 0$ satisfies $x^2 + y^2 + z^2 = 1$ I would like to ask what is the maximum number of ...
0
votes
1answer
27 views

Relationship between 2 Dimensional Quadratic systems and roots

Given four points $(x_1, y_1) (x_2, y_2) (x_3, y_3) (x_4, y_4)$ How does one construct a system of two equations: $a_1x + a_2x^2 + a_3y + a_4y^2 + a_5xy = c_1$ $b_1x + b_2x^2 + b_3y + b_4y^2 + ...
2
votes
0answers
190 views

Can a linear combination of even Legendre polynomials have common real root(s) with a linear combination of odd Legendre polynomials?

I am using the following definition of Legendre Polynomials: $P_0(x)=1$, $P_1(x)=x$ and $$P_{k+1}(x)=\left(\frac{2k+1}{k+1}\right)xP_k(x)−\left(\frac{k}{k+1}\right)P_{k−1}(x)$$ Let ...
9
votes
4answers
976 views

Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

If $a,b,c$ are non-negative numbers and $a+b+c=3$, prove that: $$\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$$ Here's what I've tried: Using Cauchy-Schawrz I proved that: $$(3a + ...
1
vote
2answers
330 views

Solve a quadratic matrix equation?

Given a known symmetric matrix $M$, vector $\vec{v}$ and scalars $a$ and $b$, I'm trying to solve for a scalar $x$ such that: $\vec{v}^T(M+(ax+b)I)^{-1}\vec{v} - ...
5
votes
2answers
2k views

Find the roots of a polynomial using its companion matrix

I would like to find the roots of a polynomial using its companion matrix. The polynomial is ${p(x) = x^4-10x^2+9}$ The companion matrix $M$ is $M={\left[ \begin{array}{cccc} 0 & 0 & 0 ...
0
votes
3answers
53 views

Determining a polynomial from its crossings with another polynomial

Is it possible to determine the coefficients of two polynomials, if we are given 2n different points at which they cross each other ? In other words, If $f(p) = \sum_{i}\alpha_{i}p^{i}$ and $g(p) = ...
25
votes
2answers
660 views

Countability of the zero set of a real polynomial

This is the question from my calculus homework: Is it possible for a polynomial $f\colon\, \mathbb{R}^{n}\to \mathbb{R}$ to have a countable zero-set $f^{-1}(\{0\})$? (By countable I mean countably ...
2
votes
1answer
123 views

Finding the Zeros of a Matrix-Vector Equation

Here's another matrix algebra question, sorry if I'm coming at these incorrectly, but this kind of thing really isn't my forte :( Lets say we have the equation: $0 = -2 \mathbf{u}^{T} \mathbf{Z} ...
3
votes
1answer
319 views

solving a univariate equation with a sum of exponentials

I am interested in a method to find the roots of the following equation: \begin{equation} f(t) = \sum_{i=1}^n \alpha_i e^{\beta_i t} + \gamma t + \delta = 0. \end{equation} For my application, ...