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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If for a polynomial $P(k) = 2^k$ for $k = 0, 1, . . . , n$, what is $P(n+1)$?

For a polynomial $P(x)$ of degree $n$, $P(k) = 2^k$ for $k = 0, 1, 2, . . . , n$. Find $P(n+1)$. If $n=1$, $P(x)=x+1$ and $P(2)=3$. If $n=2$, $P(x)=0.5x^2+0.5x+1$ and $P(3)=7$. How to approach ...
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0answers
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Bourbaki - Algebra Chapter IV - Section 6, Exercise 9(b)

Let $S_i(X_1,\dots,X_n)$ be the elementary symmetric functions in the variables $X_1,\dots,X_n$. Let $r_1,r_2,\dots,r_n$ be $n$ rational functions in the $X_1,\dots,X_n$. Let $T$ be a variable ...
2
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1answer
45 views

Algebraic and transcendental functions over $\mathbb Z_n$ — is this a known result?

I have proved a result that seems (to me) interesting, and I am wondering whether it is a known result, and if not whether it seems interesting to others. The result is as follows: Let $R=\mathbb Z ...
0
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1answer
13 views

Determinant with one parameter, how to deal with this?

Let $t\in \mathbb R$ be a parameter, and $$|A(t)|= \begin{vmatrix} a_{11}+t &a_{12}+t &\cdots &a_{1n}+t\\ a_{21}+t &a_{22}+t &\cdots &a_{2n}+t\\ \vdots &\vdots ...
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0answers
23 views

Prove there are infinitely many polynomials $P$ such that $P<CH(P)$.

Prove there exists a constant $C(n) > 0$ such that for any $ξ ∈ [0, 1]$ there exists infinitely many polynomials $P(x) = a_nx^n + · · · + a_0 ∈ \mathbb{Z}[x]$ such that $|P(ξ)| < ...
0
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1answer
33 views

Is the derivative of a exponential function a^x always greater than the derivative of a polynomial x^n as x approaches infinity

with n and a being any constants > than 1. I have tried taking the $\lim\limits_{x \to \infty} a^x / x^n$, and l'hopitals is telling me than $x^n$ can always be reduced to 1 with multiple iterations, ...
0
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1answer
14 views

Polynomial approximation on affine varieties

Let $V,W \subseteq \mathbb{A}^n$ be two affine varieties over an algebraically closed field $k$ of characteristic zero and let $a,b\in k$. Q: Can we find a polynomial $f \in k[X_1,...,X_n]$ such ...
0
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0answers
37 views

Chebyshev representation of polynomial

In Carl de Boor's A Practical Guide to Splines (1978) problem II.3.a demands a proof that a polynomial $P_ng$ of order $n$ which agrees with a function $g:\mathbb{R}\rightarrow\mathbb{R}$ at $\tau_1, ...
0
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2answers
29 views

$x^{p^{k}-1}-1$ divides $x^{p^{n}-1}-1$ in $\mathbb{F}_{p}$ iff $k$ divides $n$

Let's consider two polynomials in $\mathbb{F}_{p}[x]$: $f(x)=x^{p^{n}-1}-1$ and $g(x)=x^{p^{k}-1}-1$. How to prove that $g(x) \mid f(x)$ iff $k \mid n$? For instance, it's worth trying this: ...
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2answers
24 views

Prove equation $(ad-bc)(a-c)^2 = (b-d)^3$, if polynomials has common root

$$\begin{split} W(x) &= x^3 + ax + b \wedge a,b \in \mathbb{R} &\wedge \mathbb{D}_W &= \mathbb{R}\\ G(x) &= x^3 + cx + d \wedge c,d \in \mathbb{R} &\wedge \mathbb{D}_G ...
4
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2answers
308 views

Factorising a cubic equation

Factorise $9x-x^3$ completely. It's simple but I'm never seem to get it right; I've got $(x-1)(-x+9)x$.
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1answer
17 views

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 ...
0
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0answers
16 views

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
31 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
63 views

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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0answers
18 views

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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0answers
74 views

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
36 views

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 ...
26
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2answers
460 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
27 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
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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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3answers
29 views

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
24 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
45 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
50 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
59 views

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 ...
0
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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
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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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0answers
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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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0answers
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 ...
3
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2answers
69 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
45 views

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
12 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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0answers
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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 ...
0
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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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+50

Is the given binomial sum almost everywhere negative as $K\to\infty$?

The binomial sum is as follows: $$\mathcal {L}^K(\theta)= \sum_{i=\lceil{K/2}\rceil}^K \binom{K}{i}\theta^i\left((1-\theta)^{K-i}-\frac{1}{2}(1-\theta)^{-K}(1-2\theta)^{K-i}\right)$$ which can also ...
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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
28 views

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
39 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 ...
2
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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 ...
0
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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 ...
1
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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
20 views

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
20 views

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, ...
0
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0answers
13 views

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 ...
0
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0answers
26 views

Series Expansion from Polynomial w/ Coefficients [closed]

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 ...