Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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3answers
101 views

Does $s(0) = s(1)$ define a vector subspace in $\mathbb C[X]$?

I believe that $s(0) = s(1)$ does not define a vector space in $\mathbb C[X]$, but I am unsure how to show it. I know it doesn't satisfy the zero vector condition, nor is it closed under vector ...
5
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1answer
76 views

Analyzing a fourth degree polynomial

Let $a,b$ and $c$ be real numbers. Then prove that the fourth degree polynomial in $x$ $acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$ has either 4 real roots or 4 complex roots. I have never solved a ...
1
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1answer
260 views

Using the Multinomial Theorem to Calculate a Finite Sum raised to an exponent

I know it's a simple question, but I keep getting different general formulas for the coefficients when I am trying to use the multinomial theorem for the following: $$ ...
2
votes
2answers
35 views

Find the value of $k$ such that $p(x)= kx^3 + 4x^2 + 3x - 4$ and $q(x)= x^3 - 4x + k$ , leave the same remainder when divided by $(x – 3)$.

$p(x)= kx^3 + 4x^2 + 3x - 4$ and $q(x)= x^3 - 4x + k$ , leave the same remainder when divided by $(x – 3)$. (a) -1 (b) 1 (c) 2 (d) -2 I am getting the value of k: $-17/29$ after equating the ...
1
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2answers
57 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
1
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2answers
69 views

Determine whether S is a subspace of P3. Vector space of all real polynomials.

ATTEMPT: Have given a small attempt just really confused on how to approach. So I got the general equation of $p(x)= a + bx +cx^2 +dx^3$. So we find the derivative? and find the values of ...
1
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0answers
51 views

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$?

How are the general skills for complete factorization of arbitrary homogeneous order polynomials in $\mathbb{C}$ ? For example: $1.$ $a^2+b^2+c^2$ $2.$ $a^2+b^2-c^2$ $3.$ $a^2+b^2+c^2+d^2$ $4.$ ...
2
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2answers
60 views

Does there exist a polynomial function for every n points, whose extremas are these points?

Given $ n $ points in $ \mathbb{R}^2 $, does there exist a polynomial function of any degree, whose extremas include these $ n $ points? Given 3 points: $ P_1 = (0,4), P_2 = (2,2), P_3 = (4,7) $ And ...
3
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1answer
64 views

a polynomial about continuous function

Let $\{a_i(x):\mathbb{R}\rightarrow \mathbb{C}\}$ be continuous functions, does there exist some continuous functions $\{\lambda_i(x)\}$ such that $$a_{n-1}(x) y^n+a_{n-2}(x) y^{n-1}+\cdots ...
0
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2answers
43 views

Conditions of the system of equations.

Find m to the equation:$$\left\{ \begin{array}{l}2x^3-\left(y+2\right)x^2+xy=m\,\,(1)\\x^2+x-y=1-2m\,\,(2) \end{array} \right.$$have experience My try: From $(1)$ and $(2)\,\Rightarrow $: ...
0
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1answer
21 views

Seeing complex roots on the graph of a polynomial

When I sketch the graph for a general second degree polynomial $y = ax^2 + bx + c$ it is easy to "see" its roots by looking at the points where $y=0$. This is true also for any $n$-degree polynomial. ...
2
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1answer
60 views

Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $ (\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
4
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0answers
78 views

All roots of a polynomial lie on a circle.

I'm stuck in the following problem and I need your help to solve it. Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that: $A_0(x) = 1; \\ ...
1
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1answer
30 views

Linear algebra, polynomial problem

Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :( Let $S$ be the following subset of the vector space $P_3$ of all real ...
0
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1answer
51 views

If f is 0 at enough points, it is the 0 polynomial?

Let $f \in \mathbb{C}[x_1,...,x_n]$, and let d be the largest $x_i$-degree of f for $0 \leq i \leq n$. Prove that f is the zero polynomial, if $f(a_1,...,a_n)=0$ for all points $(a_1,...,a_n) \in ...
0
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1answer
28 views

If $abx^2 = (a-b)^2 (x+1)$ then $ [1 +(4/x)+(4/x^2)]^{(1/2) }=$?

As the title says. I found this question in our next term's book. A) (a+b)/(a-b) B) (a-b)/(a+b) C) a^2 +ab D) none
0
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0answers
38 views

Solving a septic equation

Solve the septic equtions \begin{eqnarray*} 0 &=&c_{2}^{2}x^{7}+c_{3}c_{2}^{2}x^{6}+\left( -c_{1}^{2}+2c_{1}c_{3}c_{4}c_{2}\right) x^{5}+\left( 2c_{3}^{2}c_{1}c_{4}c_{2}-c_{3}c_{1}^{2}\right) ...
4
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1answer
82 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
0
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1answer
39 views

Roots for quintic equations

I have been pondering over this question for a few months now. Why exactly do quintic equations have no close general expression for their roots? Looking at graphs and reading about it hasn't really ...
2
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2answers
114 views

Help in this proof in Lang's Algebra book

I'm trying to understand this part of the proof: I didn't understand why not all coefficients of $f_2,\ldots,f_n$ can lie in the maximal ideal, maybe I'm forgetting something, it should be a very ...
1
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1answer
25 views

Linear algebra, question about polynoms

A,B are matrices n*n over a field F. I am given a polynom f(t) {belongs to F[t]} . How can I show that Af(BA)B= ABf(AB)? I defined a polynom g(t)= t*f(t). Then I substituted AB instead of t, but I ...
9
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1answer
233 views

$\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$

Could any one tell me how to solve this one? Given $f\in C[0,\infty)$ such that $f(x)\to 0$ as $x\to\infty$ we need to show that for any $\epsilon>0$ there is a polynomial $p$ such that ...
4
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6answers
156 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
0
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2answers
52 views

Curve Fitting a Cyclical Pattern of Data

I'm analyzing phonological characteristics of the 22 letters used in the Hebrew alphabet, and assigned each letter an enumeration to see if they are organized based on place of articulation: ...
2
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0answers
24 views

Using the discriminant find the real solutions?

I have an equation: $$x^4+ax^3−b^2$$ for which the discriminant is $$−b^4(256b^2+27a^4)$$ If $$b≠0$$ what are the 2 real solutions to the equation? For these two solutions, what is a=?
5
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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
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1answer
33 views

Legendre Polynomial Orthogonality and Size

Show $(P_i,P_j)=\begin{cases} 0& i \neq j \\ \frac{2}{2j+1} & i = j\end{cases}$ for $0 \leq i, j\leq2$ I'm just not sure exactly what I'm supposed to do. Do I plug in values of i and j and ...
2
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1answer
33 views

Factorising a complex polynomial over C

If $f(z)=z^3+7z^2+16z+10$, find all factors of $f(z)$ over $C$. If I had at least one zero or factor I would be able to find the others, but I just don't know how to start.
0
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2answers
15 views

Counting polynomials which are constant on a specific hyperplane

Say a function $f:\mathbb{R}^m\to\mathbb{R}$ is having the 1-property if it equals $1$ on the hyperplane $\sum_{i=1}^m x_i=1$. How many polynomials of degree $n$ have the 1-property?
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2answers
54 views

Determine whether this is a subspace of $P_3$

Let $S$ be the following subset of the vector space $P_3$ of all real polynomials $p$ of degree at most 3: $$S=\{p\in P_3\mid p(1)=0, p^\prime (1)=0\}$$ where $p^\prime$ is the derivative of $p$. ...
0
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1answer
43 views

Bounds on coefficients of close polynomials

I've got two polynomials $p, \hat{p}:\mathbb{R}^2\rightarrow \mathbb{R}$ of degree $2\times2\ $ which are close together around $0$: $$|p(\mathbf{x})-\hat{p}(\mathbf{x})|<\varepsilon \quad \forall ...
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4answers
40 views

How do I work with equations with more than two variables?

I was trying to rewrite this equation in terms of $s$: $$ p = 4s \frac{(s - 1)}{2} + s (2r + 1) $$ After failing at that, I tried with Wolfram Alpha, and got the answer I wanted. But, how did it get ...
1
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3answers
206 views

Show $f(X)=a_nX^n+\cdots+a_1X+a_0$ has degree $n$ modulo $N$, $f(a)\equiv 0$ (mod $N$) then $f(X) \equiv (X-a)g(X) $(mod $N$)

In Niels Lauritzen, Concrete Abstract Algebra, I'm having trouble showing the following: The problem starts out like this: $f(X)=a_nX^n+\cdots+a_1X+a_0, a_i \in \mathbb Z, n \in \mathbb N$ Part ...
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0answers
29 views

Express symmetric polynomial $\prod_{i < j} (X_i+X_j)$ in terms of elementary symmetric functions

Exercise: Define a polynomial $\Sigma(X_1,\ldots,X_n)$ as \begin{align*} \Sigma(X_1,\ldots,X_n) = \prod_{i < j} (X_i+X_j) \end{align*} This is a symmetric polynomial, quite clearly. I want to ...
1
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2answers
43 views

Remainder of the polynomial

A polynomial function $f(x)$ with real coefficients leaves the remainder $15$ when divided by $x-3$, and the remainder $2x+1$ when divided by $(x-1)^2$. Then the remainder when $f(x)$ is divided by ...
3
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1answer
21 views

Generic points as coefficients of polynomial kernels?

I am reading the paper Dual-to-Kernel Learning with Ideals. Here is part of it: The definition/motivation of genericity in Wikipedia are A generic point of the topological space $X$ is a point ...
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1answer
56 views

Working out the discriminant to a polynomial and using for working out “a”

For an equation: $$ x-b^2/x^3+a=0 \\$$ i.e. $$ x^4-b^2+ax^3=0 \\$$ If the discriminant is positive (i.e. $> or =0$) for real roots, what is the discriminant for these equations? Can you use the ...
3
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2answers
430 views

Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros

I need a proof to show that the inequality $m < n$ leads to a contradiction and $P_n$ has $n$ distinct real roots, all of which lie in the open interval $(-1, 1)$.
3
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2answers
52 views

$f$ has root $\alpha$, then $f = (X-\alpha)g$ for some $g$

I need some help with the following problem: Suppose $R$ is a unique factorisation domain and $f \in R[X]$ such that $\deg f > 0$ and $f$ has a root $\alpha \in R$. Then $f = (X-\alpha)g$ for some ...
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1answer
46 views

Solutions for $a$ by factoring a multivariate polynomial

I have an equation: $$\left(\frac{b}{x^2}+1\right)⋅\left(x−\frac bx\right)+a=0$$ The question is by factorizing what are the solutions for a? I am not sure how to do this: I have reduced the ...
1
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0answers
20 views

Show equivalence corresponding Nulls of function.

I'd like to show that the following two propositions are equivalent: (1) $f \in \mathbb{R}[x]$ has a multiple Null, so it's $\ge 2$. (2) $f$ and $f'$ have a common Null, whereas $f'$ describes the ...
0
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2answers
14 views

Estimate the error of interpolating ,,,

Estimate the error of interpolating (${lnx}$) . at ${x=3}$ with an interpolation polynomial with base points ${x=1 , x=2 , x=4 , x=6 }$ .
2
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3answers
101 views

Find the polynomial ${P(z)}$ of degree ${3}$ such that …

I meant that if we have ${P(z)}$ of degree ${3}$ such that ..... $${P(-1)=7} , {P(2)=3} ,{P(4)=-2} ,{P(6)=8}$$ Find the polynomial
1
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3answers
93 views

How to factor the polynomial $x^4-x^2-2x-1$?

By taking advantage of a computer algebra system, I found out that $$x^4 - x^2 - 2x - 1 = (x^2 + x + 1)(x^2 - x - 1)$$ However, I don't know a straightforward way to solve this by hand. This was in ...
4
votes
2answers
91 views

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real?

Can you find a Polynomial of Degree 7 that has 2 complex roots and 5 real? The polynomial, call it $f(x)$ must be irreducible over $\mathbb{Q}$ (or over $\mathbb{Z}$ as Gauss' lemma can be used.) ...
0
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1answer
47 views

Can the 3-dimensional Cauchy inequality be certified by 2 squares?

The Lagrange identity writes $(a^2+b^2+c^2)(x^2+y^2+z^2) - (ax+ by+cz)^2$ as a sum of 3 squares of real polynomials, namely $[(ay-bx)]^2 + [(az-cx)]^2 + [(bz-cy)]^2$. Do 2 squares suffice?
0
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1answer
26 views

minimal polynomial and linear transformation

If $T:\Bbb{C} \to \Bbb{C}$ defined by $T(x)=x$ . T satisfity minimal poly is $x-1$. Is it correct. Any polynomial of degree $>1$ is a linear transformation on C .this type of transformation exist ...
10
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4answers
162 views

Coefficients of a polynomial also are the roots of the polynomial?

How many real solutions $(r_1, r_2, \cdots, r_n)$ are there such that $(r_1, r_2, \cdots, r_n)$ are the roots of the polynomials $x^{n} + r_1 x^{n-1} + r_2 x^{n-2} + \cdots + r_n$ For $n = 2, 3, 4$ I ...
0
votes
1answer
29 views

Uniqueness of infinite polynomial functions

This is not a homework question, it is just something I was wondering about. Suppose we have 2 sequences of real numbers, ${a_i}$ and ${b_i}$, and their respective polynomials $A(x) = ...
4
votes
1answer
66 views

Show that $f(z)=\sum_{n= 0}^{+\infty}a_n z^n$ is a polynomial

Let $f(z)=\sum_{n= 0}^{+\infty}a_n z^n$, the radius of convergence $\ge 1$. For all $n,\quad a_n\in \mathbb{Z}$ and $f$ is bounded the open unit disk. Show that $f$ is a polynomial. My ...