For questions about finding factors of e.g. integers or polynomials
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How do I show this polinomial is irreducible over $\mathbb{Q}$? [duplicate]
I'm trying to show that $P(x) = \frac{x^{p^2}-1}{x-1}=x^{p^2-1}+\cdots +1$ is irreducible over the rationals, as $p$ an odd prime. Here it is my try: As $P$ is a primitive polinomial, it's enough to ... abstract-algebra factoring irreducible-polynomials- 33
find all integers m such that the number $m^5+m+1$ is prime
I really don't know how to start this problem. The problem should be related to factoring the equation to derive an answer, but I don't know how. factoring rational-functions- 1
Scary integral function [duplicate]
If $f,g,h,\phi$ are polynomials in $x$, and $$p(x)= \left (\int_1^xf(t)h(t)dt\right) \left(\int_1^xg(t)\phi(t)dt\right)-\left(\int_1^xf(t)\phi(t)dt\right) \left(\int_1^xg(t)h(t)dt\right) $$ is ... calculus polynomials definite-integrals factoring- 731
Please check if my solution is correct [duplicate]
Let $f(x)=a x^{2}+b x+c$ be a quadratic polynomial with integral coefficients, where $a \neq 0$. Show that (i) if $f(x)$ is factorisable into linear factors with integral coefficients, then there are ... algebra-precalculus polynomials solution-verification factoring- 73
How to rearrange equation to cubic polynomial form?
Summarize the problem I have these 2 equations and i need a cubic polynomial in terms of Z so i can run numerical routines on it. How can i be sure that's possible to factor that way and second how ... algebra-precalculus polynomials systems-of-equations factoring- 1
Factor $a x^n + b x + c = 0$
I wish to factor the polynomial equation $$a x^n + b x + c = 0$$ When $a = b$ and $n=5$, we have the Bring-Jerrard normal form of the quintic $x^5 + x + c = 0$. Using the Lagrange Inversion theorem, I ... polynomials factoring- 494
$(x-\lambda)$ and $(x-\overline{\lambda})$ appear the same number of times in the factorization of $p$ Linear Algebra Done Right 3rd Edition Axler
I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 4.15 Polynomials with real coefficients have zeros in pairs Suppose $p\in\mathcal{P}(\mathbb{C})$ is a polynomial with ... polynomials factoring- 6,235
Is the polynomial $p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$?
My initial question in the present post is pretty basic: Is the (integer) polynomial $$p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$? When $k=... algebra-precalculus number-theory polynomials factoring square-numbers- 10.2k
Assume that $a=b$. If the function $f(x)$ is monotonously increasing, then (answer 1) $0<a<1$ (answer b).
The given function is $f(x)=x^3-3ax^2+3bx-2$. I am aware that monotonously increasing means to continuously increase, so I tried getting this function's derivative and then setting it to zero, but to ... calculus algebra-precalculus quadratics factoring- 137
Factoring $1 + 2\cos(\theta_1) x + 4i\sin(\theta_1) \cos(\theta_2) x^2 + x^3=0$
Is it possible to factorize the cubic $$1 + 2\cos(\theta_1) x + 4i\sin(\theta_1) \cos(\theta_2) x^2 + x^3=0$$ without explicitly using the cubic equation? Given the coefficients, the equation will ... polynomials factoring- 494
Other ways to factorize $xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$
To factorize $xy\left(x+y\right)+yz\left(y-z\right)-xz\left(x+z\right)$ I used the fact that $x=-y$ and $y=z$ and $x=-z$ make the expression zero. Hence it factorize to $\lambda (x+y)(y-z)(x+z)$ and ... algebra-precalculus factoring- 5,637
Factoring $m^4+2m^3-m^2-2m-8$
Factorize $$m^4+2m^3-m^2-2m-8$$ First I plugged numbers like $m=-3,-2,-1,0,1,2,3$ in the expression and neither of them is a root so I can't use rational roots theorem here to find a factor. I've ... algebra-precalculus factoring- 5,637
How to find factors of a matrix determinant
I'm working on some problems for fun and came across this problem I'm stuck with. Here's the question: Use row operations to show that $ x + \omega y + \omega^2 z $ is a factor of $ \Delta $, where $ \... linear-algebra determinant factoring roots-of-unity- 31
Find the intersection point between these two equations
We have $$ f(x) = 12\sqrt x $$ and $$ g(x) = x^2 - 7x + 12 $$ I need to find where they intersect. So far I've reduced the expression to $$ 12\sqrt x = x^2 - 7x + 12 $$ $$ 12 \sqrt x = (x-4)(x-3) $$ ... polynomials factoring- 125
Can a number have infinite number of factors if we include rational numbers
Can a number such as 10 have infinite factors if we include multiplying two rational numbers or a rational number and integer or any other combinations? Google says factors of 10 are 1×10, 2×5 and the ... factoring- 5
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