1
vote
1answer
53 views

I need help proving a statement about rational roots

I have no idea where to start...this is the statement: If a polynomial of degree not greater than 5 with rational coefficients has multiple roots, it has also a rational root, except in the case ...
0
votes
6answers
62 views

solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
0
votes
3answers
59 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
1
vote
0answers
17 views

Descartes rule of signs extension

Let $V(\text{sequence})$ be the number of sign changes in the sequence, e.g. $V(-3,0,-2,9,0,1)=1$. Show that $V(a_0,a_1,...,a_n)\ge V(a_0,a_0+a_1,a_0+a_1+a_2,...)$. Furthermore, prove that if ...
0
votes
1answer
33 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
0
votes
5answers
67 views

Graphing polynomials

Sketch a graph of the polynomial $P(x)=(x-2)^2(x+1)^3$. You must plot and label the x and y intercepts and these should be the only points you plot. How do I sketch the graph of a polynomial?
1
vote
2answers
47 views

how to solve this question of polynomials

Given the polynmial is exactly divided by $x+1$, when it is divided by $3x-1$, the remainder is $4$. The polynomial leaves remainder $hx+k$ when divided by $3x^2+2x-1$. Find $h$ and $k$. This ...
0
votes
2answers
46 views

how to find the remainder when a polynomial $p(x)$ is divided my another polynomial $q(x)$

i was solving the question from the book IIT FOUNDATION AND OLYMPIAD - X and i was solving the problems of polynomials-III. so ...
0
votes
3answers
70 views

Roots of $x^2 +2x +2$ Over $\mathbb{C}$

Find the roots of $x^2 +2x +2$ over $\mathbb{C}$ I need to prove somehow that the roots will be $(1 + i) , (1 - i)$ Any ideas how can I find those roots in a simple way?
2
votes
2answers
124 views

help interpreting an abstract algebra test question

This is a take-home test problem, and I don't want help solving it, just understanding what it's asking. I've asked my prof a couple times, but she's either unwilling or unable to give me a straight ...
2
votes
1answer
41 views

Irreducible factor decomposition

This is a past exam question. Decompose each of the following elements as a product of irreducible: (a) $X^4+2 \in \mathbb{Z}_5[X]$ (b) $X^5+X \in \mathbb{Z}_2[X]$ (c) $X^5+4X^4-3X^3+X^2+7X+11 \in ...
0
votes
1answer
29 views

Irreducible in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{R}[X]$ and $\mathbb{C}[X]$

Is the polynomial $2X^3-10X^2+50X+10$ rrreducible in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{R}[X]$ and $\mathbb{C}[X]$. What I have done so far is, since this is a cubic function, therefore it will ...
0
votes
3answers
40 views

Explanation on characterstic polynomial

$A_2 = \begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix} $ So the characteristic polynomial of $A_2$ is $P_a(t) = (t-1)^2 - a $ Then, $ P_a(t) = t^2 -2t +1 -a$ ...
1
vote
1answer
35 views

GCD of polynomials over $\mathbb{Z}_3$

$f$ and $g$ are polynomials over field $\space \mathbb{Z}_3$. $f=X^4+X^3+X+2, \space g=X^4+2X^3+2X+2$. And I been asked to find the GCD of them. What I have done is using Euclidean algorithm. After ...
0
votes
2answers
57 views

Please answer my question :D Factorisation of cubic expressions.

Given that the remainder when $x^3-x^2+ax$ is divided by $x+a$, where $a > 0$, is twice the remainder when it is divided by $x-2a$, find the value of $a$.
2
votes
1answer
80 views

Proof that a is an eigen value of p(T) if and only if a=p(lambda) for some eigenvalue lambda of T

$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\C}{\mathbb{C}} \newcommand{\LM}{\mathcal{L}}$ Question: Suppose $\F = \C, T \in \LM(V), p \in ...
0
votes
3answers
59 views

Primitive Root in Quotient Ring

Find a primitive root of $R[x]/\langle x^4+x+2 \rangle$ where $R$ is the integers mod $3$. Is there a good general stratagy to this sort of thing?
9
votes
1answer
131 views

Galois group of the quintic polynomial $X^5+X+1$

I'm trying to find the Galois group of the polynomial $p(X)= X^5+X+1$ over $\mathbb Q$. First, one notes that, if $\omega$ is a primitive cubic root of unity, then it is a root of $p(X)$. So, ...
0
votes
1answer
53 views

Writing a piecewise polynomial function as a sum of truncated power functions

Writing a piecewise polynomial function f(t) as a sum of truncated power functions p(t)= (t-c)^k where f(t) is defined as $$ f(t) = \begin{cases} 0 &, 0 \leq t < 1 \\ t - 1 ...
3
votes
1answer
28 views

Existence of a certain polynomial in $\mathbb Z [X,Y]$

I am at a point where I need to know whether there is a polynomial $f \in \mathbb Z [X,Y]$ such that: $f(1,y) \ge 0$ for all $y \ge 0$ $y-1,x \ge 0 \wedge f(x,y) \ge 0 \Rightarrow 0 \le f(2x,y-1) ...
4
votes
4answers
183 views

Prove that p has m distinct roots if and only if p and p' have no roots in common

Problem: Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p'$ have no roots in common. My proof so far: If $m=0$, ...
1
vote
1answer
64 views

Exercise about basis

I am trying to solve the following exercise: Let $A \in L(P_3)$ be defined by a matrix: where $A^b_e$ = $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$ ...
0
votes
1answer
40 views

Finding the value of $n$ in this question?

Find the value of n for which the quadratic equation $$ \sum_{k=1}^{n}(x+k-1)(x+k) =10n $$ has solutions $α$ and $α+1$ for some $α$.
0
votes
3answers
94 views

Quotient rings of polynomial rings

I have come across a quite difficult question while I am studying for a test: Let $F=\Bbb Z[x]/(7,x^2-3)$. Let $u$ denote the image of $x$ under the canonical epimorphism from $\Bbb Z[x]$ to ...
1
vote
2answers
42 views

Remainder theorem question

If $n$ is an integer, what is the remainder when $$3x^{2n+3}-4x^{2n+2}+5x^{2n+1}-8$$ is divided by $x+1$? How would we know what the value of $n$ is?
0
votes
0answers
30 views

How to determine if a basis is top-down?

I am unclear as to what the rules are that define a set to be a top-down basis. For example: what are top-down basis for P2? Below is the grid of polynomials for P2. \begin{array}{ccc} (t-1)^2 ...
0
votes
2answers
55 views

$x^3+3x^2+4x+5=0$ and $x^3+2x^2+7x+3=0$, how many common roots they have?

My attempt, Equate both, at the end you will get $x^2-3x-2=0$ That means $x=-1$ and $x=2$. But what after that. Please provide solutions as well.
5
votes
0answers
57 views

Minimising an expression - involving polynomial

I found this one on a forum but it has been unanswered from long there. I am curious to know if there is a solution to this problem. Here it is: Let n be a positive integer. Determine the smallest ...
1
vote
0answers
34 views

Matlab: Calculate Lagrange interpolaton polynomial

I'm new to Matlab so I don't even know where to start. I need to calculate Lagrange's interpolation polynomial of some function f in the case when the nodes are ...
0
votes
1answer
26 views

$(x+b)^3\mid P(x)+a$ and $(x-a)^3\mid P(x)-a$

$a,b\in\mathbb{C}$, $b!=0$ I need to find all the polynomials $P$ of degree $5$ verifying: $ \begin{cases}(x+b)^3\mid P(x)+a\\ (x-b)^3\mid P(x)-a\end{cases} $ PS : there was en error, i fixed it ...
0
votes
2answers
32 views

Greatest common divisor of polynomials over $\mathbb{Q}$

I have two polynomials: $f: x^3 + 2x^2 - 2x -1$ and $g: x^3 - 4x^2 + x + 2$. I have to do two things: find $gcd(f,g)$ and find polynomials $a,b$ such as: $gcd(f,g) = a \cdot f + b \cdot v$. I have ...
0
votes
3answers
76 views

Find all values of $a$ for which there are two real solutions of $x^3-2ax^2+a^2x-3=0$

Find all values of $a$ for which there are two real solutions of the equation. $$x^3-2ax^2+a^2x-3=0$$ Ans = $1.5\sqrt[3]{6}$ I tried to research the function by dint of derivative, but it didn't ...
1
vote
2answers
84 views

how to find the all zeros of ploynomial $x-\sqrt 5$

Given that $x-\sqrt 5 $ is a factor of the cubic polynomial $x^3-3\sqrt 5x^2+13x-3\sqrt 5$, find all the values of the polynomial after the long division method i get $x^2-2\sqrt 5x+3$ now ...
0
votes
1answer
53 views

How to find the value of $a$ and $b$ in polynomials

Here is a question from NCERT-Exampler pg-15 question no. 6 For what value of $a$ and $b$, are the zeros of $q(x)=x^3+2x^2+a$ also the zeros of polynomial $px(x)=x^5-x^4-4x^3+3x^2+3x+b$ ? Which ...
2
votes
1answer
33 views

Problem related to Cyclotomic Polynomials

I'm trying to show that if $p$ is prime, then $$x^{p-1}-x^{p-2}+x^{p-3}-...-x+1$$ is irreducible over $\mathbb{Q}$. I don't have an idea of how to start. I know the $p^{th}$ cyclotomic polynomial is ...
0
votes
2answers
28 views

how to solve this question of polynomial

if $(x-3)$ and $x-1/3$ are both the factors of polynomial $f(x)=ax^2+5x+b$, then (a) $a=5b$, (b) $a=b$, (c) $a=3, b=2$, (d) $a=+-b$ to solve this question i kept $x-3$ equal to $0$ and ...
0
votes
1answer
34 views

how to find the value of K in quadratic equations

look at the following question If one of the Zeros of polynomial $f(x)=Kx^2-17K+(3K-2)$ is reciprocal of other zero, then K is equal to: (a) 1, (b) -1, (c) 2, (d)-2 i solve the above ...
-4
votes
2answers
118 views

Select the approximate values of x that are solutions to $f(x) = 0$, where $f(x) = -7x^2 + 6x + 9$? [closed]

These are the answer choices: $$\begin{align*}\\ A&~~\{–0.78, 0.67\}\\ B&~~\{-7, 6\}\\ C&~~\{–0.86, –1.29\}\\ D&~~\{–0.78, 1.64\} \end{align*} $$
0
votes
3answers
31 views

Parameter “m” for which $P(x)=4(m+1)x^3+(m-3)x+1-m$ has a root with multiplicity two…

Can you please help me solve this parametric problem. So, we have to find all the values of real parameter $m$ for which the following equation has a solution with multiplicity ...
4
votes
1answer
29 views

Finding the conjugates in $\mathbb{C}$ of a given number over a given field…

I'm having somewhat of a difficult time understand what's being asked—and thus having a hard time answering the question: Find all conjugates in $\mathbb{C}$ of the given number over the given ...
1
vote
3answers
27 views

Roots of Polynomial Equation?

$y=1/x$ so I plugged in $x=1/y$ into the equation above and got $y^{4}+y^{3}+y^{2}/c+y/4-1/2$, but apparently it's wrong, when I looked up the answer below. What am I missing?
1
vote
2answers
22 views

Size of an ideal in a polynomial Ring

Let $F$ be a field and let $I = \{f(x) \in F[x]\mid f(a) = 0 ~~ \forall a \in F\}$. Prove that $I = \{0\}$ when $F$ is infinite. I have already shown that $I$ is an ideal and that $I$ is infinite ...
5
votes
3answers
1k views

Give the remainder of $x^{100}$ divided by $(x-2)(x-1)$.

What will be the remainder obtained when the polynomial $x^{100}$ is divided by the polynomial $(x-2)(x-1)$. I used remainder theorem but it had no impact in the solution.
2
votes
3answers
42 views

Prove that for $\forall x,y,z \in \mathbb{R}, x^2+y^2+z^2\geq xy+yz+xz $

Show that for $\forall x,y,z \in \mathbb{R},x^2+y^2+z^2\geq xy+yz+xz $. I first assumed that $x\geq y \geq z$, but I'm having problems with the $z^2$. By itself, $z^2$ is clearly not greater than ...
13
votes
3answers
738 views

Solve $3n^3 + 3n^2 + 4n = n^n$ in positive integers

So my cousin is in the math team (7th grade) and he was asking me for help on one of his problems but I don't know how to solve For what positive integer n does $3n^3 + 3n^2 + 4n = n^n$ anyone know ...
1
vote
3answers
92 views

If $(1 + 2i)$ and $(3 - 2i)$ are two roots of $x^5 + ax^4 + bx^3 + cx^2 + dx + 4$, then $a$ =?

Consider the polynomial $x^5 + ax^4 + bx^3 + cx^2 + dx + 4$ where $a, b, c, d$ are real numbers. If $(1 + 2i)$ and $(3 - 2i)$ are two roots of this polynomial then what is the value of a? Well, I ...
0
votes
0answers
22 views

Descartes rule of signs

I'm trying to write an algorithm that gets a polynomial and gives how many roots does it have in the interval [0 $x_0$]. I'm supposed to do it by Descartes law, I know that by Descartes law you Know ...
6
votes
4answers
134 views

If a polynomial $g$ divides $f$ and $f'$, then $g^2$ divides $f$?

Here's a homework problem from Artin's Algebra that I'm having a lot of trouble with Let $f(x) \in F[x]$ (where $F$ is a field of characteristic $0$). If $g$ is an irreducible polynomial that is ...
0
votes
3answers
37 views

Euclidean algorithm in the ring of polynomials over a field

I need some help with the following division proofs. I suppose my biggest problem is not being able to visualize the algebra for one GCD equaling another GCD. I'm not sure of how to arrange the ...
0
votes
4answers
59 views

Proof that the coefficients of a polynomial are real

How does one prove that all the coefficients of this polynomial: $$(x+i)^{10}+(x-i)^{10}$$ are real numbers, without using Newton's Binomial Theorem?