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.

learn more… | top users | synonyms

1
vote
0answers
27 views

How can I find $f^{ad}$ along with $\operatorname{Im} f^{ad}$ and $\operatorname{Null} f^{ad}$?

Let $ f \in \mathbb R_{\leq4}[t]$ such that $f(p)=f \bigg( \sum\limits_{k=0}^4 a_k t^k\bigg):=f \bigg( \sum\limits_{k=1}^4 k a_k t^{k-1}\bigg)$ and define $\langle p,q \rangle := \int\limits_{-1}^1 ...
10
votes
3answers
327 views

How prove this Polynomial $g(x)=\sum_{i=1}^{n}a^m_{i}x^i$have only real roots?

Question 1: let Polynomial $f(x)=\displaystyle\sum_{i=0}^{3}a_{i}x^i,$ have three real numbers roots,where $a_{i}>0,i=1,2,3$. show that: $$g(x)=\sum_{i=0}^{3}a^m_{i}x^i$$ have only real ...
2
votes
1answer
60 views

Eigenvalue of $A$ is root of $p(t)$

I'm working on this proof problem: I have part (a) done. I think this is the Cayley Hamilton Theorem, so I got some insight from Wikipedia. I am a little confused with part (b) though because I ...
0
votes
2answers
652 views

Finding polynomal function with given zeros and one zero is a square root

I've been having trouble with this problem: Find a polynomial function of minimum degree with $-1$ and $1-\sqrt{3}$ as zeros. Function must have integer coefficients. When I tried it, I got this: ...
5
votes
6answers
4k views

Why can't polynomials have negative exponents or division by a variable

Why can't: $$2x^{-3} - 3x$$ or $$\frac{1}{2x}$$ be polynomials too? Why have a definition that excludes these algebraic forms?
1
vote
0answers
79 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 ...
3
votes
0answers
388 views

Convert CRC to result of reversed polynomial?

Looking at the Wikipedia page for CRCs I see that they list a bunch of standard CRC polynomials along with the Reversed Polynomials of each. If I have a value that was calculated with a certain ...
0
votes
1answer
44 views

Determine the interpolating polynomial

Determine the polynomial of $ deg \le 6 $ interpolating function $$ f(x) = x^3 + 2x^2 + x + 1 $$ at the points : $ -3, -2, -1, 0, 1, 2, 3 $. My first idea it was to use Lagrange's formula, but it's ...
0
votes
2answers
561 views

Prove that this polynomial is irreducible over $\mathbb Z$

I want to prove that the following polynomial is irreducible: $$x^3 - x^2 - x + 3$$ My question gives the hint to apply the substitution $x \mapsto x+1$ but I've tried this and when multiplied out I'm ...
1
vote
1answer
168 views

On irreducible polynomial over normal extension

Let $L/K$ be a normal extension and a irreducible polynomial $f(X) \in K[X]$. Prove that, if $f$ is reducible over $L$ then $f$ is factored into product of irreducible factors with same degree. ...
0
votes
1answer
416 views

Factor into a product of irreducible polynomials

Since the polynomial $p=x^4−2$ is irreducible over $\mathbb{Q}$, the factor ring $K=\mathbb{Q}[x]/(p)$ is a field. I'd like to factor the polynomial $q=y^4−2$ in $K[y]$ into a product of irreducible ...
0
votes
2answers
147 views

Cubic with repeated roots has a linear factor

If $f$ is a cubic polynomial with a repeated root over a field then $f$ has a linear factor. I think that is true in perfect fields but I don't know how to prove it.
1
vote
1answer
213 views

Hermite polynomials form complete system

Let $h_0(x)=e^{-x^2/2}$ and $h_k=B^kh_0$, where $B=-\dfrac{d}{dx}+x$. Show that the $\dfrac{h_k}{\|h_k\|_2}$'s form a complete orthogonal system. We can show that $h_k(x)=H_k(x)e^{-x^2/2}$, ...
1
vote
1answer
70 views

A question about cubic roots of rational numbers

I'm trying to understand if, given $K$ a cubic cyclic extension of $\mathbb{Q}(\zeta_3)$, where $\zeta_3$ is a third primitive root of unity, it always exists $b \in \mathbb{Q}$ such that $\sqrt[3]{b} ...
1
vote
2answers
215 views

On polynomial of prime degree.

Let $K$ be a field, $f(X)\in K[X]$ be a polynomial of prime degree. Assume that for all extension $L$ of $K$, if $f$ has roots in $L$ then $f$ splits over $L$. Prove that either $f$ is irreducible ...
1
vote
2answers
41 views

What can we say about $\dim \operatorname{null}(AB)$ from knowing $p_A$ and $p_B$?

Say, there are two matrices $A, B \in \mathbb R^{3,3} $ such that their characteristic polynomials are $p_A(t) = t^3 − t^2 + 2t$ and $p_B(t) = t^3 − 7t^2 + 9t − 3$. What do we know about $\dim ...
4
votes
1answer
123 views

Given the polynomial $(x-1)(x-8)(x-31)-1$, how do you conclude that its roots are irrational?

Example $(x-1)(x-8)(x-31)-1$. Just by looking at this polynomial how do you conclude that the roots are irrational?
4
votes
1answer
71 views

How prove this polynomial such $\dfrac{1}{f(x_{i})}=h(x_{i}),i=1,2,\cdots,s$.

Question: let polynomial $\phi{(x)}$ is an irreducible polynomials on the rational number field.and assume that $x_{1},x_{2},\cdots,x_{s}$ is $\phi{(x)}$ complex roots, for any $f(x)$ is rational ...
0
votes
1answer
41 views

Number of monic polynomials of degree $2$

I know that there are $p^2$ monic polynomials of degree $2$ over the field $\mathbb{Z}_{p} $ but I want to prove it precisely. Help me a hint to prove that. Thanks a lot.
5
votes
1answer
98 views

Complete orthogonal system from polynomials

Let $h_0(x)=e^{-x^2/2}$ and $h_k=B^kh_0$, where $B=-\dfrac{d}{dx}+x$. Show that the $\dfrac{h_k}{\|h_k\|_2}$'s form a complete orthogonal system. (Hint: We have $\langle Af,g\rangle=\langle ...
0
votes
1answer
66 views

Prove that the roots of $2x^3 - x + 5 = 0$ are irrational

We want to prove that for the equation $2x^3 -x + 5 = 0$, any root must be irrational. How can this be done? Seems like plugging in $x = a/b$ doesn't really help at all.
1
vote
1answer
366 views

Newton backward interpolation in Mathematica

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + ...
1
vote
1answer
170 views

Non-commutative indeterminates in polynomial rings.

Described below are some observations I have made while fiddling around with polynomials. In addition to the two questions below, I am looking for any sort of relevant information so I can read more. ...
1
vote
0answers
98 views

Orthogonality of the Hermite polynomials: probabilistic approach

Can anyone help me with the following question: Is there any reference in which a probabilistic approach was used to prove that the Hermite polynomials are orthogonal? Thanks a lot!
2
votes
1answer
504 views

Fourier transform on Hermite polynomial

Let $h_0(x)=e^{-x^2/2}$ and $h_k=B^kh_0$, where $B=-\dfrac{d}{dx}+x$. Define a transformation $T$ as $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$ How can I find the ...
2
votes
3answers
114 views

$\gcd$ of polynomials over a field

I have the polynomials $f,g\neq 0 $ over a field $F$. We know also that $\gcd(f,g)=1$ and $$ \det \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\neq 0. $$ I ...
1
vote
0answers
42 views

Show a polynomial is reducible to linear terms - check my answer

I have an exam tomorrow in linear algebra, and I want to make sure I answered this question correctly. Let $p \in \mathbb R[x], z \in \mathbb{C}$. We are given if $Im(z)>0$ then $p(z)\neq0$ Show ...
0
votes
1answer
65 views

Solving an equation in a field.

I need to know the way to solve equations like this: $$(x^2+1)f(x) = 1 \pmod{x^3+1}$$ over a field $F_{3}[x]$. Thanks in advance for any help.
0
votes
2answers
62 views

Prove that there are rational polynomials $p,q$ such that $p(x)(x^4+2x^2+1)+ q(x)(x^4-3x^2-4) = x^2+1$

Prove that there are polynomials $p(x), q(x)$ in $\mathbb{Q}[X]$ such that: $$p(x)(x^4+2x^2+1)+ q(x)(x^4-3x^2-4) = x^2+1$$ Is it still true if we replace $x^2+1$ with $x+5 $? So: I ...
2
votes
1answer
866 views

History of polynomial arithmetic

How did the notions of polynomial addition,multiplication and division develop historically? The fact that this correspondence with the integers exists seems to be of great importance and is not at ...
4
votes
3answers
134 views

$x^5-1$ completely splits in $\mathbb F_{16}$

I need to prove that $x^5-1$ completely splits in $\mathbb F_{16}$. This means it has exactly $5$ unique roots in $\mathbb F_{16}$. I have only found the following way: find an irreducible polynomial ...
1
vote
1answer
23 views

Univariate Representation of Affine Transformation

Definition: Let $0\leq i \leq n$ and $A_i,B_i \in \mathbb{E}$. Then we call the polynomial $S(X)=\sum_{i=0}^{n-1}B_i X^{{q}^i}+A$ the univariate representation of the affine transformation $S(X)$. ...
1
vote
1answer
116 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 ...
2
votes
1answer
58 views

Is this a homogeneous polynomial

If $f(ta,tb) = f(a,b)$ $\forall t \neq 0$, then can we conclude that $f$ is a homogeneous polynomial of degree 1?
7
votes
1answer
165 views

$x^n - a$ is irreducible over $\mathbb{Q}$?

Let $a$ be a positive rational number and $n$ be a positive integer such that $\sqrt[k]{a} \notin \mathbb{Q}$ for $k=2,3,\ldots,n$. It is true that the polynomial $x^n - a$ is irreducible over ...
0
votes
2answers
140 views

Finding the quadratic coefficient of a quartic polynomial given other coefficients.

This is a question which I want to solve, taken from this sample question paper for an exam I'm appearing for tommorow: The product of two of the roots of $$x^4-11x^3+kx^2+269x-2001$$ is $-69$. ...
0
votes
2answers
66 views

Linear transformation of polynomials, given transformation output

I need to find a transformation of a polynomial, given the output of other polynomial calculations: If $T : P_1 \mapsto P_2$ is a linear transformation such that, $$ T(1 + 5x) = 1 - 2x ...
2
votes
1answer
91 views

If $X$ is a cone, show that $I(X)$ is homogeneous.

The exercise is 1.3(3) from HP Kraft, "Appendix A: Basics from Algebraic Geometry." If a closed subset $X\subseteq \mathbb C^n$ is a cone, show that $I(X)$ is generated by homogeneous functions. ...
1
vote
0answers
55 views

Divisors and cyclotomic polynomials

Let $n \in \mathbb{N}^{\ast}$ and $\Phi_{n}(X)$ be the $n$-th cyclotomic polynomial defined by : $$ \Phi_{n}(X) = \prod \limits_{\substack{1 \leq k \leq n-1 \\ \gcd(k,n)=1}} \Big( X - \exp \big( ...
0
votes
1answer
89 views

Find the multiplicative inverse of $\,x^2+(x^3-x+2)$ in the quotient $\,F_3[x]/(x^3-x+2)$

Find the multiplicative inverse of $x^2+(x^3-x+2)$ in the quotient $F_3[x]/(x^3-x+2)$ . I've proved that $x^3-x+2$ is irreducible polynomial in $F_3[x]$, and that $x^2$ and $x^3-x+2$ are coprime ...
1
vote
0answers
41 views

Fixing learning rate for gradient descent single variable

I need guaranteed convergence to local minimum given initial value in $(0,6)$. The function is $f(x) = 30-61 x+41 x^2-11 x^3+x^4$. I have $x(i+1) = x_i - \eta (4x_i^3-33x_i^2+82x_i - 61) $. What ...
4
votes
3answers
186 views

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$.

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$. Well, if $\deg P\le 3$ this is easy since we can deduce $P(0)=P(1)=P(-1)$ by letting $x=0,1,-1$
5
votes
2answers
119 views

What's the most basic yet interesting Algebraic Geometry result regarding this polynomial?

Let $f(x,y,z) = x^a + y^b - z^c$, where $a,b,c \gt 0$. What is the most basic yet interesting result about this polynomial from Algebraic Geometry?
1
vote
2answers
87 views

Inverse of polynomial over $\mathbb F_3$ finite field, quotient space

A question about quotient spaces, something I do not fully understand yet, and can use some help. $A = \mathbb F_3[x]$, $P = x^3-x+2\in A$ 1) Show that $P$ is irreducible (I did it, it has no roots ...
1
vote
1answer
59 views

Factoring a polynomial over finite field $\,F_3$ that has a root

A question I am struggling with. We are asked to factor $\,f(x)=x^2+x+1$ over the field $F_3 =\{0,1,2\}$ So, I checked for a root, and I saw that $f(1) = 1^2+1+1 =0$ (because $3=0$ in $F_3$) that ...
1
vote
2answers
94 views

quadratic equation with two variables

i try to solve the equation below has a solution or not $x^2-97y-40 =0$ if solution exists, $x^2-40$ must be congruent to 0 modulo $97$. if i could show the congruence above implies that ...
2
votes
0answers
47 views

Congruence of polynomials

Prove that $y^i \equiv y^{i \pmod 4} \pmod {y^4 + 1}$ for all $i \ge 0$. Any hints on how to solve this? I don't even know where to start.
1
vote
1answer
88 views

Integer polynomial, maximum number of consecutive integer values that it can reach.

Lets say I have an integer polynomial $P(x)$ of degree $n$ and $x_0,\dots,x_r \in \mathbb{Z}$ such that $P(x_0) = 0$ $P(x_1) = 1$ $P(x_2) = 2$ $\dots$ $P(x_r) = r$ What is the largest $r$ that I ...
0
votes
1answer
35 views

How to express $\sum_{i=0}^{n}a_i^m$ in terms of $\sum_{i=0}^{n}a_i$ and $\sum_{i=0}^{n}a_i^2$

Is there any way to express $\sum_{i=0}^{n}a_i^m$ by polynomial of $X=\sum_{i=0}^{n}a_i$ and $Y=\sum_{i=0}^{n}a_i^2$? For example, if $n=2$ and $m=3$, it can be expressed as $\dfrac{X}{2}(X^2+Y)$. I ...
2
votes
1answer
134 views

Does this higher-order polynomial have an analytic solution?

I know that in general polynomials above degree 4 do not have analytic solutions, except for a few special cases. What I want to know is whether this particular polynomial is one of those cases. The ...