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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expansion of polynomials of higher degrees

How to expand $(x-x_n)(x-x_{n-1})...(x-x_0)$ into $a_nx^n+...+a_0$? Surely, $a_n=1$ is equal to one in my case, but how to find out the rest of coefficients? Do we a numerical algorithm of calculating ...
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0answers
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Effect on existing roots of polynomial when adding small higher-order term

How do existing roots of a polynomial change when adding higher-order term with a small coefficient? Given a sufficiently small coefficient of the new higher-order term, the existing roots shouldn't ...
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2answers
27 views

Example such that $ f_1(x)$ is reducible but $f(x)$ is irreducible.

(The (mod p) Irreducibility Test) Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by ...
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3answers
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Prove that for any real number the inequality is true: $x^4-x^3+5x^2 > 3x - 6$

Prove that for any real number the inequality is true: $x^4-x^3+5x^2 > 3x - 6$ The only way I could do this is to transform this inequality to: $x^4-x^3+5x^2-3x+6> 0$ and then sketch the ...
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How do I solve massive system of equations (with lots of variables) quickly?

Just wondering how to solve system of equations involving 3+ unknowns quickly. In my math class, we're given questions like these which involve solving huge system of equations on a time limit, ...
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Integral Inequality $\leq n^{3/2}\pi$

$ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real roots. Show that: $ \int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$ It's easy to see ...
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1answer
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Newton's sum help

How would one solve the following system with Newton's sums and Vieta's relations?: $$x+y+z=14$$ $$x^2+y^2+z^2=14$$ $$x^3+y^3+z^3=34$$ I have taken an algebra lesson in the awesome math academy, but ...
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2answers
367 views

Only 12 polynomials exist with given properties

Prove that there are only 12 polynomials that have all real roots and whose coefficients are $-1$ or $1$. Zero coefficients are not allowed, and constant polynomials do not count. Two of them ...
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0answers
23 views

Proving a version of maximum modulus principle elementarly.

There is this version of maximum modulus principle: If $P$ is a non-constant polynomial, then $|P|$ doesn't have a local maximum. I know that if $P$ is non-constant, then $|P(z)| ...
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3answers
98 views

Find the coefficient of $x^{30}$.

Find the coefficient of $x^{30}$ in the given polynomial $$ \left(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}\right)^5 $$ I don't know how to solve problems with such high degree.
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What is that function? Polynomial?

Is it a polynomial or rational polynomial or else? $y = \dfrac{a}{x^4} + \dfrac {b}{x^2} + c$ I need to fit a curve to a discrete data of that form, so I need to know what fitting to use.
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0answers
22 views

coefficients of polynomial and binomial expressions

Let us say we are given a polynomial p(x)=$\sum_k a_k x^k$. In order to find $\sum_k a_k$ we simply need to evaluate p(1), and similarly there are many other tricks. Is there any trick to evaluate ...
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1answer
43 views

An approach to proving that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$

I have to prove that $\Bbb{Q}[x,y]/(x^3-y^2)$ is isomorphic to $\Bbb{Q}[t^2,t^3]$. My approach: Let us consider $t^2$ and $t^3$ as separate variables $x$ and $y$. The relations that hold for them ...
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3answers
48 views

Prove $\lim_\limits{x\to\infty}\dfrac{P_k(x)}{P_{k+1}(x)}=0$ [on hold]

Prove $\lim_\limits{x\to\infty}\dfrac{P_k(x)}{P_{k+1}(x)}=0$ by limits. $P_k(x)$ is defined as a polynomial of degree $k$.
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1answer
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Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$?

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$? I tried Faulhaber's formula and Bell number but couldn't ...
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3answers
20 views

Consider the equation [x^5+x=10] show that i)the equation has only 1 real root ii)this root lies between 1 & 2 iii)the root must be irrational [on hold]

This equation obviously has 5 roots..If they are considered as a,b,c,d,e then a+b+c+d+e=0,abcde=-10..but what next?If i procced through contradiction will it help?
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0answers
27 views

Given $f=x^4+x+1 \in \mathbb Z_{2}[x]$ is primitive, write down an $m$-sequence ${a_n}$ associated to $f$

I'm not sure how to solve this question exactly. I know that the period will be 15 but I don't know how to construct the $m$-sequence.
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Solving quartic equation? (Cardano/Ferrari)

today I've written a little Code-Snippet that is based upon the steps that are mentionned in this wikipedia-Article to solve a general quartic polynom. Here's my matlab-implementation: ...
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91 views

Cardinomials: Like cardinalities, but polynomial valued

I want to see if this notion is known (or if it makes sense). Let $F$ be a field. Let $A$ be a finite dimensional commutative unital algebra over $F$. Let $X_1$, $X_2 \in A$ etc. be such that their ...
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1answer
36 views

general theorem on roots of a polynomial needed to show it's identically zero.

Polynomial degree k, one variable, if it's zero at k+1 values, then it's identically zero. Can someone point me to a proof of this? I know derivatives at points can count as these roots (if k-degree ...
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2answers
65 views

Prove: if $a$ and $b$ are algebraic, then $a + b$, $a - b$ and ab are also algebraic

I have to prove the following: If $a, b \in \mathbb{C}$ and are both algebraic over $\mathbb{Z}$, then: $a + b$ is algebraic over $\mathbb{Z}$ $a - b$ is algebraic over $\mathbb{Z}$ $ab$ is ...
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5answers
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Cubic Equation. (Factorisation)

I'm given this question, factorise $4x^3-7x-3$. Is this answer acceptable? $(x+\frac{1}{2})(x-\frac{3}{2})(x+1)$.
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1answer
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Find $n$ such that $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$.

I have to find the form of n i.e. whether n is even or odd and whether it is multiple of 2 or 3 such that: $x^2 + x + 1$ is a factor of $(x+1)^n - x^n - 1$. What I tried: $x^2 + x + 1 = (x + 1)^2 - ...
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1answer
11 views

Solving the Cubic Equation (using Lagrange Resolvents)

This is from my textbook. I am having trouble working out the calculations that the author skips over. So we start with the polynomial $\ X^3 - aX^2 + bX -c$ with zeros $x_1,x_2,x_3$. Then we define ...
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1answer
11 views

How to find representation of polynomial w.r.t different basis

Let $B$ be the basis of the vector space of polynomials of degree less than or equal to 2. $B = \{1, t-1,(t-1)^2\}$. Let $u = 2t^2-5t+6$. How do you find $u_b$, the coordinate vector of $u$ relative ...
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$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th symmetric polynomial of $n$-variable. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...
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1answer
20 views

if $f(n+1)-f(n)=P(n)$, exist a polynomial $Q(x)$ such that for all $n \in \mathbb{Z}$ : $Q(n)=f(n)$

Let $f:\mathbb{Z} \to \mathbb{Z}$ such that, exist a polynomial $P(x)$: $$f(n+1)-f(n)=P(n)$$ for all $n \in \mathbb{Z}$ Prove that exist a polynomial $Q(x)$ such that for all $n \in ...
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0answers
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Recursive relationship for Peano Baker Series

The Peano Baker Series is a integral has the following form $$\varPhi(h,0)=I+\intop_0^h G(t_{1}) \, dt_1 + \intop_0^h G(t_1) \intop_0^{t_{1}} G(t_2) \, dt_2 \, dt_1 + \intop_0^h G(t_1) ...
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0answers
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Is the given binomial sum almost everywhere negative as $K\to\infty$?

The binomial sum is as follows: $$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$ It can be found ...
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seeking for Newton's like inequalities as sufficient condition for polynomial to have only real zeros

For polynomial $P_n(x)=\sum_{k=0}^n a_k x^k, a_k>0$, it is known that a necessary condition for $P_n(x)$ to have only real zeros is that Newton's inequality holds: ...
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1answer
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An obstacle encountered in a proof of the existence of a best approximating polynomial of degree $\leq n$

Let $n \in \{0, 1, 2, \dots\}$, let $a, b \in \mathbb{R}$ be such that $a < b$ and let $f \in \mathcal{C}[a, b]$ be a real function that is continuous on the non-degenerate, compact interval $[a, ...
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1answer
38 views

Equation that defines multi-dimensional polynomial

In two-dimensions a complete n-th degree polynomial is given by $P_n(x,y) = \sum_{k=0}^{n}\alpha_kx^iy^j \qquad i+j \leq k \qquad (1)$ . However, now I am dealing with the following two-dimensional ...
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1answer
66 views

Ordered Pairs of Polynomials

Professor proposed this problem to the class today. Suppose we had $P_1(x), P_2(x) \in \mathbb{Z[x]}$, $n, a \in \mathbb{Z}$. How many ordered pairs exist such that ...
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1answer
40 views

I have to show $p=p(x-\lambda)$ if and only if they have the same zeros in $F$

Suppose $F$ is a field, $|F|\geq n \geq 2$. Given $\lambda \in F$ I know $p,p(x-\lambda)\in F[x]$ are irreducible monic polynomials. I have to show $p=p(x-\lambda)$ if and only if they have the same ...
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1answer
49 views

Taylor polynomial for an integral

This is the first time encountering a Taylor expansion along with an integral, so I am wondering how I should proceed. Question: $Consider \space the \space function$ $$F(x) = ...
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0answers
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Zero Homogeneous Polynomials

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a homogeneous polynomial of degree n. Is it true that if $\forall x,f(x)=0$, then the coefficients of $f$ are all zero?
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1answer
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Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$.

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$. I know there are lots of answers you could write, but would this be correct: $\{(x-2)^5, (x-2)^4, ...
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Polynomial Division for crc

I did this question by just using the xor long division of the binary, but my teacher said he doesn't want it done that way, but want me to use polynomial Division. I have no clue how to do this, and ...
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Series Expansion from Polynomial w/ Coefficients [on hold]

I have four coefficients to a 4-the order polynomial. Besides having some stroke of luck finding a pattern (that would be difficult considering the coefficient values) what is the best way to approach ...
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1answer
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Find all polynomials such that $P(A)\subset U$ for a countable subset of the unit circle $U$

I recently answered a question, in which I proved that If a polynomial fixes the unit circle then $P$ is a monomial (a classical result),i,e: $$\forall P\in \Bbb C[X]\ \ \ \ (\forall z\in \Bbb C \ \ ...
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1answer
109 views
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Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
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1answer
30 views

If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det > B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the ...
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Writing a series of polynomial equations of certain degree from a sequence of binary bits using Magma

How do I write a series of polynomial equations of a specified degree from a sequence of binary bits using Magma. So far, I have the following code for converting a decimal sequence to binary. ...
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2answers
55 views

Polynomials mod prime $p$

The problem is $5m^2+m+4 \equiv 0\pmod 7$. I am supposed to first convert it to a quadratic whose first coefficient is $1$. But the polynomial cannot be factored, so I am unsure as to how to do ...
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Let $P(x),Q(x) \in \mathbb{Z}[x]$ such that, exist $a,b \in \mathbb{Z}^+$ and $a<b$: $P(a)=Q(a)$ and $P(b)=Q(b)$

Let $P(x),Q(x) \in \mathbb{Z}[x]$ such that, exist $a,b \in \mathbb{Z}^+$ and $a<b$: $P(a)=Q(a)$ and $P(b)=Q(b)$ Prove that $P \equiv Q$
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1answer
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For any $n \in \mathbb{Z^+}$ Not extis $P(x) \in \mathbb{R}[x]$ with coefficients in $B$ and all roots of $P(x)$ in $A$

Problem: Let $A=\{a_1,a_2,..,a_m\}$ and $B=\{b_1,b_2,...,b_p\}$ where $a_1,a_2,...,a_m,b_1,b_2,...,b_p \in \mathbb{R}$ Prove that , the following statements is bad : for any $n \in ...
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1answer
54 views

Help required! Polynomials

Let $D(p) = p^{20} - p^{18} - p^{16} - \dots - p^2 - 2$ Prove that the sum of fourth powers of all the real roots of $D(p) = 8.$ Please help.
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1answer
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Limit at $\infty$ of a polynomial multiplied by a negative exponential

I am trying to show $\int_0^{\infty} x^2 e^{-2 x} dx = 1/4 $ Integration by parts gets the indefinite integral $$\int x^2 e^{-2 x} dx = \frac{-1}{4} e^{-2 x} (2 x^2+2 x+1)+constant$$ In order to ...
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How can I find the closure of $P[a,b]$ [closed]

Let $P[a,b]$ the space of all polynomials on the interval $[a,b]$ clearly $P[a,b]$ is a subspace of $C[a,b]$ but how can find the closure of $P[a,b]$ , In special case $[0,1]$ .
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Vector subspace. [closed]

$H = \lbrace p(x) \in P_2 \vert p(1) = 0 \rbrace $ is a vector subspace of $P_2$. What is a basis for for $H$ and the $\dim (H)$? I think the dimension is $0$ since th restriction of p(1)=0, is that ...