All Questions
14
questions
-1
votes
2
answers
71
views
Is a polynomial monotone when the first derivative has only imaginary roots? [closed]
I have a polynomial over a specific the range. The first derivative has only two imaginary roots and no real roots. The first derivative is positive in the lower bound and upper bound. Does that mean ...
- 9
0
votes
0
answers
86
views
How can you find the roots of a polynomial if all the roots are imaginary?
I am trying to find the roots of a 4th-degree polynomial by hand which is actually coming from the derivative of the open-loop transfer function of a control system to find the breakaway points for ...
- 123
0
votes
0
answers
82
views
Prove that the polynomial $P(x) = x^{p−1} + 2x^{p−2} + 3x^{p−3} +···+ (p − 1)x + p$ is irreducible in $\mathbb Z[x].$ [duplicate]
Let $p$ be a prime number. Prove that the polynomial
$P(x) = x^{p−1} + 2x^{p−2} + 3x^{p−3} +···+ (p − 1)x + p$
is irreducible in $\mathbb Z[x].$
The Eisenstein-Schönemann theorem cannot be used to ...
- 615
0
votes
0
answers
153
views
Proof? of irreducibility of cyclotomic polynomial
Let $\omega = e^{\frac{i2\pi}{n}}$.
I am trying to show that the minimal polynomial of $\omega$ over $\mathbb{Q}$ is the cyclotomic polynomial, that is the polynomial whose roots are the primitve $\...
- 3,623
6
votes
1
answer
847
views
$x^a+ y^b + z^c$ is irreducible in $\mathbb C[x,y,z]$
Let $a,b,c$ be positive integers. Then $f = x^a + y^b + z^c$ is irreducible in $\mathbb{C}[x,y,z]$.
By Gauss, $f$ is irreducible in $\mathbb{C}[x,y,z]$ iff is so in $\mathbb{C}(z)[x,y]$, and so iff ...
- 1,662
1
vote
1
answer
139
views
Irreducible fatorization of $X^n-1$ in both the ring of polynomials with complex coefficients and real coefficients
Let $f(x) = x^n-1$ be a polynomial in $\Bbb R[x]$. Factorize $f(x)$ as a product of irreducible polynomials in $\Bbb C[x]$ and show that if $n$ is even, $f(x)$ has two reals roots and if $n$ is odd, $...
- 482
1
vote
1
answer
429
views
Roots of polynomial with imaginary coefficients
This is the first time I see this kind of problem, so it might be trivial but I am just not used to it.
What are the roots of $x^3-6ix^2-11x+6i$
I am not sure If I should ignore the imaginary ...
- 375
0
votes
2
answers
244
views
How to Factor this polynomial as a product of irreducible polynomial over complex
I want to factor this polynomial as a product of irreducible polynomial in $\mathbb{C}$
$f(x)=X^n - 1 \in \mathbb{R}[X]$
But I’m not sure how to do it. I know that $x^2 -1 is (x-1)(x+1)$ and that the ...
- 135
0
votes
1
answer
589
views
Express polynomial as product of irreducible factors over fields
I've seen some similar questions around but I'll post this up anyways. There's this beautiful polynomial: $$z^4-z^3-5z^2-z-6$$ and I am to factorise it into a irreducible polynomial in $\Bbb Q$, $\Bbb ...
1
vote
2
answers
478
views
How to find the roots for $x^3+2x+2$
I seem to be forgetting my college algebra. Can someone help me understand how to find the complex roots of the polynomial: $x^3+2x+2=0$
I tried synthetic devision for the possible real roots of $\...
- 27
0
votes
1
answer
260
views
How to apply Eisenstein's criterion in complex polynomial
I have two questions
First,
If $f(x) = x + i$, how can I apply Eisenstein's criterion to prove that $f(x)$ is irreducible over $\mathbb{C}[x]$?
Second,
For $f(x, y) = x^2 + y^3$, $f(x,y)$ is ...
- 5
2
votes
3
answers
3k
views
Evaluating $2x^3+2x^2-7x+72$ where $x= \frac{3-5i}{2} $
So I have to find the value of :
$$2x^3+2x^2-7x+72$$ at $$x= \frac{3-5i}{2} $$
Where $i$ stands for $\sqrt{-1}$.
I know the obvious approach would be to substitute the value of $x$ in the equation, ...
- 89
1
vote
1
answer
106
views
Find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$
I want to find all complex roots of $T^4-{1/2}T^2-\sqrt{15}T+{69/16}$.
The only way I can think to do it is to find 1 complex root, $\alpha$, by inspection, so we can rearrange the polynomial to be $...
- 2,179
1
vote
0
answers
50
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 ...
- 12.8k