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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93 views

Rewrite $\frac{1}{1-\sqrt[3]{2}}$ as a polynomial question

I've been looking for a way to rewrite the following fraction as a polynomial equation in $\sqrt[3]{2}$: $$\frac{1}{1-\sqrt[3]{2}}.$$ Now, to rewrite $1/(1-\sqrt{2})$ as a polynomial equation, it is ...
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0answers
65 views

Diagonalization of a linear transformation in the polynomial vector space

Let $V = R_3[X]$ be the vector space of polynomials with real coefficients of degree at most 3 and consider the linear transformation $V \rightarrow V$ defined by $f_a(p(x))=p(1-ax)$ for each $p(x) ...
0
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1answer
44 views

Find a counter-example for inequality

Let $C>1$ be a constant. I have to find polynomial $p(t)=a_0+a_1 t+\dots +a_n t^n$ such that: $$|a_0|+|a_1|+\dots + |a_n| \le C \sup_{t\in[0,1]} |p(t)|$$ doesn't hold. Any tip?
4
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2answers
40 views

Find all the polynomials $p \in \mathbb R [X]$ such that $(x+1)p=(p')^2$

(Where $p'(x)$ is the derivative of $p(x)$) Research effort: what I thought is that given that $(x+1)|(p')^2$ then $(x+1)|(p')$ (I'd like to justify better this, but I don't know how) Then, ...
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2answers
153 views

How prove this here exsit $b\in R$,such $S=\{(b,b,\cdots,b)\}$,if $f(x_{1},x_{2},\cdots,x_{n})$ is the set of minimum and maximum points.

Assmue $f(x_{1},x_{2},\cdots,x_{n})$ is a second degree real polynomial with $n(n\ge 2)$ variables. Let $S$ be such that $f(x_{1},x_{2},\cdots,x_{n})$ is the set of minimum and maximum points. In ...
3
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1answer
190 views

Monic irreducible polynomial over an integral domain

These days, I have some basic problem in abstract algebra. I know that in any integral domain, any prime element must be an irreducible element. Moreover, if $A$ is a UFD, then an element $a \in A$ is ...
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2answers
121 views

Polynomial Solutions for Differential Equations

Suppose we have a set of polynomials where $\deg(Q_k(x))\le k$, and consider the following differential equation, $$W:=\sum_{k=0}^n Q_k(x)\frac{d^k}{dx^k} .$$ It is known that if there is a ...
0
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1answer
28 views

irreducibility of $x^2-a$ in $\mathbb{Z}_2[a]$

Let $a$ be transcendental over $\mathbb{Z}_2$ and let $F=\mathbb{Z}_2(a)$. Prove $p(x)=x^2-a$ is irreducible over $F$ I've been trying to understand this for a while now, but I'm ...
0
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1answer
84 views

Leading coefficient of a polynomial quotient is equal

Isn't there something special that you can infer when a polynomial's leading coefficients are equal on the numerator and denominator, like this: $\frac{x^4}{x^4+3x^4}$ I'm trying to find the limit ...
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4answers
124 views

1, 3, 6, what is the next number of the sequence?

I've heard (and believed even without proof) that given any finite sequence there is more than one formula for which the same first inputs give the same first outputs. Given that: f(1)=1 f(2)=3 ...
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1answer
30 views

Lowest norm solution to a system of polynomial equations

I have a system of cubic equations: $$0=A_0+A_1 x+A_2 ( x \otimes x ) + A_3( x \otimes x \otimes x )$$ where $\dim A_0 = \dim x$ (so there are as many equations as unknowns). You may assume that the ...
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3answers
43 views

How prove this $f(x)$ and $g_{t}(x)$ be relatively prime.

Let the real coefficient polynomials $$f(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$$ $$g(x)=b_{m}x^m+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}$$ where $a_{n}b_{m}\neq 0,n\ge 1,m\ge 1$, and let ...
2
votes
1answer
164 views

Degree of a function

I found on wikipedia (http://en.wikipedia.org/wiki/Degree_of_a_polynomial) that a degree of a general function can be computed as $$\deg f(x) = \lim_{x\to\infty}\frac{\log |f(x)|}{\log x}$$ or $$\deg ...
1
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1answer
32 views

What is fixed in a equation in a polynomial vector space

From what I've learned, an equation $p(t)$ in $P_n$ is defined $$p(t) = a_0+a_1t+a_2t^2+\cdots+a_nt^n \tag 1$$ Given the basis $\beta=\{1,t,t^2,\ldots,t^n\}$, $p(t)$ can be written in the form $$p(t) ...
4
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1answer
776 views

Dimension of the vector space of homogeneous polynomials

Let $k[X_0, X_1, \ldots, X_n]_d$, or briefly $k[X]_d$, be the $k$-vector space whose elements are the zero polynomial and homogeneous polynomials of degree $d\geq 1$. I found the following formula for ...
0
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2answers
67 views

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
15
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3answers
551 views

How prove $P(x)=\sum_{k=0}^{n}(2k+1)x^k$ is irreducible over $\mathbb{Q}$

Show that the polynomial $$P(x)=\sum_{k=0}^{n}(2k+1)x^k,\forall n\in N^{+}$$ is irreducible over $Q$. My try: Since $P(x)$ has integer coefficients and the gcd of these coefficients is $1$, by ...
5
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2answers
224 views

How to solve these three equations?

If α ,β ,γ are three numbers s.t.: $\ α^ \ $ + $\ β \ $ + $ γ \ $ = −2 $\ α^2 \ $ + $\ β^2 \ $ + $ γ^2 \ $ = 6 $\ α^3 \ $ + $\ β^3 \ $ + $ γ^3 \ $ = −5, then $\ α^4 \ $ + $\ β^4 \ $ + $ ...
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2answers
974 views

Product and Sum of Polynomial Roots

The ratio of the sum of the roots of the equation, $8x^3+px^2-2x+1=0 $ to the product of the roots of the equation $5x^3+7x^3-3x+q=0 $ is $3:2$. What is the value $\frac{p-q}{p+q}$? Well I found out ...
2
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0answers
97 views

Descartes Rule of Sign for exponential sums

I have the following exponential sums ($x\in\mathbb{R}$) $$f(x)=\sum_{i=1}^Na_iP_i(x)b_i^x$$ where $P(x)$ is some monomial, e.g., $x^2, x^3,\dots$, so $f(x)$ looks like ...
3
votes
1answer
131 views

Minimize norm of a polynomial on a circle

Let $P=\sum_{k=0}^n a_kX^k$ ba a polynomial of degree $n \gt 0$, and let $r\gt 0$. Suppose that $P$ is not the monomial $a_nX^n$, in other words there is at least an $i<n$ such that $a_i\neq 0$. ...
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2answers
39 views

Separability of $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$

This is an example in Ash, Basic Abstract Algebra, ch.3.4 page 73 at the bottom (or here on page 11). It states that $f(x) = (x-1)^2(x-3)$ over $\mathbb{Q}$ is separable. But, $f'(x) = ...
5
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6answers
12k views

How do you factor $x^3-3x^2+3x-1$?

$$x^3-3x^2+3x-1?$$ I know this may seem trivial, but I, for the life of me, I cannot figure out how to factor this polynomial, I know that the root is $$(x-1)^3=0$$ because of wolframalpha, but I ...
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1answer
167 views

Computing univariate resultant via modified Euclidean algorithm

In an answer to the question Resultant of Two Univariate Polynomials, a PDF of course slides was linked which describes a modification of Euclid's algorithm for computing univariate polynomial ...
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1answer
131 views

Resultant of Two Univariate Polynomials

I am trying to implement an algorithm for computing Res(f(x),g(x),x) where f(x) and g(x) uni variate polynomials with integer coefficients. Could any one list the various algorithms for computing ...
4
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1answer
27 views

Constructing a family of polynomials in $\mathbb R_n[X]$

Let $\mathbb R_n[X]$ be the vector space of polynomials of degree at most $n$. Let $u$ the endomorphism sending $P$ to $P(X+1)-P(X)$. I want to show that there exists a unique family of polynomials ...
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0answers
80 views

Questions on polynomial ring in several variables

Let $K$ be an infinite field. Prove that different polynomials in $R=K[X_1,X_2,...,X_n]$ don't lead to the same function $K^n \to K, x \mapsto f(x)$. (solved) Find $I \subset R$ and different ...
0
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2answers
79 views

How to factor cubics having no rational roots

$$-8x^3 +8x -3 = 0$$ I've already tried the possible roots of $\pm 1$ and $3$ using the rational roots test, but none of these help break it down into something more workable. How do I solve this ...
3
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0answers
55 views

Showing that the n first derivatives of (x²-1)^n have at least r roots (for the r-th derivative)?

I have f(x) = (x²-1)^n. I want to show that, for r = 0,1,2,...,n, the r-th derivative is a polynomial (that's easy to show) that has no fewer than r distinct roots in (-1,1). I guess I need to use ...
2
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2answers
472 views

Prove, that two equations are equivalent

EDIT: Missed something very important! Sorry! We have $x^4+1=2(2x-1)^{1/4}$ not $x^4+1=2\sqrt{2x-1}$. One friend of mine told me that the equation $x^4+1=2(2x-1)^{1/4}$, where $x\geq \frac{1}{2}$ is ...
5
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6answers
728 views

What should be added to $x^4 + 2x^3 - 2x^2 + x - 1$ to make it exactly divisible by $x^2 + 2x - 3$?

I'm a ninth grader so please try to explain the answer in simple terms . I cant fully understand the explanation in my book . It just assumes that the expression that should be added has a degree of ...
8
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1answer
94 views

Non-polynomial representations of $GL_n$

Recall that every finite-dimensional rational representation of $GL_n$ is of the form $(\det)^{-k} \varrho$ for some integer $k\geq 0$ and polynomial representation $\varrho$ (and $\det$ is the ...
2
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1answer
125 views

Polynomial Problem, $P(x+1)P(x-1)=P(x^2+1)$

$P(x)$ is a real polynomial such that $P(x+1)P(x-1)=P(x^2+1)$. Find $P(x)$. I have no idea how to start on this problem. The only things I could do was finding things like $P(2)P(0)=P(2)$ or ...
3
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2answers
112 views

(Integer) Variant of Hilbert’s irreducibility theorem

Let $P\in{\mathbb Q}[X,Y]$ such that $P(x,.)$ has an integer root for any integer $x\in{\mathbb Z}$. Does it follow that $P$ has factors of the form $Y-Q(X)$ for some $Q\in{\mathbb Q}[X]$, and does ...
0
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1answer
50 views

How to simplify floor polynomial given lower bound on x?

$$ \left\lfloor\frac{8x^2 + 5x -4}{3x^2 + x}\right\rfloor $$ where $x$ > $\sqrt{8}$ How would you simplify this type of expression? *Please note the floor operation surrounding the expression ...
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1answer
47 views

positive integer polynomial under the usual polynomial multiplication

consider the set of polynomials with positive integer coefficients together with the operation, usual multiplication of polynomials. now my first question is does this set together with the ...
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2answers
106 views

Long division, explain the result!

I have this: $$ \frac{x^2}{x^2+1} $$ Wolfram Alpha suggests that I should do long division to get this: $$ 1- \frac{1}{x^2+1} $$ But I don't understand how it can be that, please explain.
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3answers
66 views

How to generically solve polynomial expressions given a minimum value of x?

Given something like $\dfrac{8x^2 + 2x + 7}{3x^2 + 2x}$, and $x > \sqrt8$, what strategy would you employ to simplify this expression?
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0answers
67 views

What are the coefficients of these “almost-Boolean” polynomials?

For any $q \in \mathbb{N}^{\ge 2}$, consider the polynomial $$ P(q) = \begin{cases} 1 + 2 \sum_{i = 1}^{\lceil q / 2 \rceil - 1} x^i + x^{q / 2} & \text{if $q$ is even}, \\ 1 + 2 \sum_{i = ...
5
votes
2answers
360 views

Is evaluation homomorphism surjective?

Let $A^n$ be an affine space over $\mathbb{C}$ and let $\mathbb{C}[X_1,\cdots,X_n]$ be the polynomial ring of $n$ variables. Then $A^n\to (\mathbb{C}[X_1,\cdots,X_n])^*$ by evaluation homomorphism, ...
3
votes
2answers
561 views

Identically zero multivariate polynomial function

Let $p$ be a prime and let $F=\mathbb{Z}/p\mathbb{Z}$. Can a nonzero multivariate polynomial $f\in F[x_1,....,x_n]$ such that $\mathrm{deg}_if< p$ for all $i=1,\ldots,n$ be identically zero as a ...
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0answers
21 views

relationship between two set of variable

i am trying to determine what kind of mathematical modeling could be applied following two variables,let us call them $x$ and $y$ ,namely change one variable has effect second on,i have several ...
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1answer
177 views

Integer coefficient polynomial - values as powers of 2

Does there exist a polynomial f with integer coefficients such that $f(0) , f(1) ... f(n) $ are all distinct powers of 2 ? I have no clue about how should i start thinking about this problem but ...
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2answers
41 views

Determine polynomials with $n$-variables

Here is a funny problem arise from harmonic analysis: Let $E$ be a measurable subset of $\mathbb R^n$ with $m(E)>0$, where $m$ is the usual Lebesgue measure on $\mathbb R^n$. In practice, $E$ ...
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2answers
59 views

Finding out the coeffcient next to $x^2$ in $(\cdots(x-2)^2-2)^2\cdots-2)^2$.

In need to find out the coefficient next to $x^2$ in polynomial $(\cdots(x-2)^2-2)^2\cdots-2)^2$, where we nest the expression $(x-2)^2$ n times. Meaning that for $n=1$ we get $(x-2)^2$, for $n=2$ we ...
0
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2answers
678 views

Suppose p(x) is a polynomial with integer coefficients. Show that if p(a) =1 for some integer a then p(x) has at most two integer roots. [duplicate]

Problem : Suppose $p(x)$ is a polynomial with integer coefficients. Show that if $p(a) =1$ for some integer $a$ then $p(x)$ has at most two integer roots. Let $p(x)$ be a cubic polynomial such ...
0
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2answers
155 views

Polynomial divisibility

Given $p(x) \in \mathbb Q[x] $ an irreducible polynomial, and $\alpha \in\mathbb C $ root of $p(x)$. Prove that if $q(x) \in \mathbb Q[x]$ it's a polynomial, such $q(\alpha) = 0$ then $p(x) \mid ...
6
votes
1answer
166 views

Homogeneous polynomials between vector spaces

Consider $\mathbb{C}$ vector space $V$ = span$(e_1,\cdots,e_n)$. Consider the following algebra embedding $$ \mathbb{C}[X_1,\cdots,X_n]\hookrightarrow F(V,\mathbb{C})$$ where $f\mapsto(\sum_i a_i ...
0
votes
1answer
60 views

if $P(P(x)))=P(x)^{16}+x^{48}+Q(x)$,Find the smallest possible degree of $Q$

let $P(x)$ and $Q(x)$ are two polynomials such that $$P(P(x)))=P(x)^{16}+x^{48}+Q(x)$$ Find the smallest possible degree of $Q$ My idea: let $$P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$$ ...
2
votes
1answer
76 views

Polynomial with degree greater than or equal to 2 not injective over algebraically closed field?

I know that this is probably a silly question. Given a polynomial in an algebraically closed field, why is the polynomial not injective if its degree is greater than or equal to 2? I understand that ...