Articles of polynomials

How can I prove that the follow polynomial is irreducible in $\mathbb{Q}$?

How can I prove that $x^5 + 7x^4 + 2x^3 + 6x^2 – x + 8$ is irrudicible in $\mathbb{Q}$? I can’t use the Eisenstein’s criterion and I tryed to put this polynomial in $\mathbb{Z}_3$ and $\mathbb{Z}_5$. Can you give me some advice please?

Explain the terms : homogeneous , symmetric , anti-symmetric , cyclic with respect to polynomials.

In answers to some of my previous questions , a lot of people used the terms homogeneous polynomial ( in a,b,c ) (under permutations of variables ) , cyclic polynomial ( in a,b,c) (under permutations of variables ) , anti-symmetric and symmetric polynomial . Please explain their meaning . Also in answering my previous question […]

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$

$B_{i}^{n}(x)={{n}\choose{i}}x^i(1-x)^{n-i}$, prove that $B_{i}^{n}(cu)=\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)$ I tried to to solve it from the right side: $\sum\limits_{j=0}^{n}B_{i}^{j}(c)B_{j}^{n}(u)=\sum\limits_{j=0}^{n}{{j}\choose{i}}c^i(1-c)^{j-i}{{n}\choose{j}}u^j(1-u)^{n-j}$. Since ${{j}\choose{i}}$ is $0$ for $j<i$, then the sum is $\sum\limits_{j=i}^{n}B_{i}^{j}(c)B_{j}^{n}(u)=\sum\limits_{j=0}^{n-i}B_{i}^{i+j}(c)B_{i+j}^{n}(u)=\sum\limits_{j=0}^{n-i}{{i+j}\choose{i}}c^{i}(1-c)^{j}{{n}\choose{i+j}}u^{j+i}(1-u)^{n-j-i}$. But I’m stuck…

Modular arithmetic with polynomial

Given n=pq, where p and q are primes, P(x) is polynomial and z∈Zn. I need to prove that: P(z) ≡ 0 mod n iff P(z mod p) ≡ 0 mod p AND P(z mod q) ≡ 0 mod q. If i could prove the more general case: P(z) mod n ≡ P(z mod p) mod […]

How to prove whether a polynomial function is even or odd

We know that a function is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$. With this reasoning is it possible to prove that a polynomial function such as $f(x) = a_{2n}x^{2n} + a_{2n-2}x^{2n-2} + …+a_{2}x^2 + a_{0}$ is even or odd? What do you suggest? How do we get started?

Can this approach to showing no positive integer solutions to $p^n = x^3 + y^3$ be generalized?

The following problem is a $2000$ Hungarian Olympiad question. Find all primes $p$ such that: $$p^n = x^3 + y^3$$ The answer is that there are only $2$ solutions: $2^1 = 1^3 + 1^3$ $3^2 = 2^3 + 1^3$ Here’s the argument: Assume $p \ge 5$ with $x,y,p,n$ positive integers and $p^n = x^3 + […]

Primitive polynomial

Prove that $x^5 + x^2 + 1$ is a primitive polynomial over ${\mathbb F}_2$. I have already proved that the above polynomial is irreducible. Do I have to exhaustively prove that the above polynomial does not divide $X^n + 1$ where $1 \le n < 31$ or is there a better way to prove this? […]

Determining whether there are solutions to the cubic polynomial equation $x^3 – x = k – k^3$ other than $x = -k$ for a given parameter $k$

Let $k$ be a real parameter, and consider the equation $$x^3 – x = k – k^3 .$$ Obviously, $x=-k$ is a solution. Is it the only one? How to prove it?

Is it possible to find function that contains every given point?

Let say we have a arbitrary number of given points and there is at least one function, for which every point lies on its graph. Is it possible to find that function using only X and Y coordinates of every given point? Example: We are given points $A(0,1)$; $B(0.27;0)$ and $C (3.73;0)$. All of these […]

Kernel of the homomorphism $\mathbb C → \mathbb C$ defined by $x→t,y→ t^{2},z→ t^{3}$.

I think we have $z-x^3$, $y-x^2$, and $z^2-y^3$ as elements of the kernel of the homomorphism $\mathbb C[x,y,z] → \mathbb C[t]$ defined by $x→t,y→ t^{2},z→ t^{3}$. But why the kernel is not generated by all the 3 elements, and only by $z-x^3$, $y-x^2$? I think maybe it is because of $z^2-y^3$ is in $\left<z-x^3, y-x^2\right>$, […]