# Tagged Questions

This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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### General method: show subset of $\mathbb{C}$ is connected

Consider the two sets $$A = \{z \in \mathbb{C} : |z^2 - 3| < 1\}, ~~~~ B = \{z \in \mathbb{C} : |z^2 - 1| < 3 \}$$ $B$ is connected, while $A$ is not. However, I have no idea how to prove this....
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### The expansion of $(a+b+c+d)^{20}$ [closed]

Let us consider the expansion of $$(a+b+c+d)^{20}.$$ Find: The coefficients of $a^{11}b^6c^2d$ and $a^{11}b^9$, The total number of terms of this expansion, The sum of all the coefficients. Thank ...
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### Prove that if $R$ is real closed field, $f \in R[X]$, and $f(a)f(b) <0$ for some $a < b$ in $R$, then $f(r) = 0$ for some $a < r < b$

Prove that if $R$ is real closed field, $f \in R[X]$, and $f(a)f(b) <0$ for some $a < b$ in $R$, then $f(r) = 0$ for some $a < r < b$. Suppose that $f$ is a irreducible polynomial of ...
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### How a complex root $\eta$ of $x^2 + x + A$ affects the ring $\mathbb{Z}[\eta]$

While reading a statement in P. Pollack's Not Always Buried Deep: A Second Course in Elementary Number Theory I came across a statement that seemed obvious and I am wondering if I am oversimplifying ...
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### Seeking an “easy ” way to show that $p(x)=x^6+\cdots+x^2+x+1$ is irreducible over $\Bbb{Z_{17}}$

As the title suggests, we need to Show that $p(x)=x^6+\cdots+x^2+x+1$ is irreducible over $\Bbb{Z_{17}}$ We can immediately answer that it is indeed irreducible since it is the cyclotomic ...
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### Why does degree determine the amount of zeros?

We just learned about complex numbers in my math class and I have a question. Why does the degree of a polynomial equal the amount of zeros it has? The degree of $f(x) = x^3 - x^2 + x - 1$ is $3$, ...
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### The ability to solve the multivariate nonlinear equations

For m nonlinear polynomial equations with n variables and the highest degree 3, how is the current ability to solve such equations? In the webpage of IBM cplex, it says that: IBM ILOG CPLEX ...
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### Proof that there are No Modulo Invertible Polynomials with f(1) = 0

I was reading an article from the NTRU Cryptosystem (probably the first one): NTRU: A Ring-Based Public Key Cryptosystem And I don't know how to prove the assertion he makes in parenthesis in ...
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### Polynomial that is irreducible over $\mathbb{Q}$ but reducible over every finite field [duplicate]

I want to prove that $X^4 - 10X^2 + 1$ is reducible in $\mathbb{F}_p[X]$ for every prime number $p$, but it is irreducible over $\mathbb{Q}$. I am not sure how to approach this problem; any ...
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### Vanishing polynomial in complex projective space

Assume we are working in $n$-dimensional complex projective space. Why does a (homogeneous) polynomial of degree less than or equal to $d - 1$ which equals $0$ on $d$ points on a line $L$ in ...
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### Showing a polynomial of degree 7 is not solvable by radicals. [closed]

Show that the polynomial of $x^7-10x^5+15x+5$ is not solvable by radicals. For a polynomial of degree 5, we simply consider the derivative and determine the number of real and complex roots from ...
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### Riesz Representation Thereom for Polynomials with real coefficients problem

Find a polynomial q(x) $\in$ P$_2$($\Bbb R$) Such that $p ( 1/4 ) = $$\int_0^1 p(x)q(x) \,dx$$$. I'm sorry to ask this question, but I've been working on it for some time. The inner product on P$_2$(...
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### Galois Group Isomorphic to $S_3$.

Let $f \in \mathbb{Q}[x]$ be an irreducible polynomial of degree 3. Suppose $f$ has one real root, we want to show that $$\text{Gal}(L/\mathbb{Q}) \cong S_3,$$ where $L$ is the splitting field of $f$. ...
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### proving an inequality involving a linear spline / piecewise polynomial

I have $n+1$ sample points $x_i = \left(\frac{i}{n}\right)^4$ and want to approximate the function $f(x)=\sqrt{x}$ by a linear spline $f_n \in S^{1,0}(\mathcal{T_n})$ on the interval $[0,1]$. I know ...
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### A problem about ring of polynomials over a field [duplicate]

For $K$ is an infinite field and $f(x_1,x_2,\ldots,x_n)$ $\in K[x_1,x_2,\ldots,x_n]$ . Prove that If $f(a)=0$ for any $a \in K^n$ then $f=0$. Can any one help me?
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### Difference between rationalizing factor and conjugate surd

I have some confusion regarding rationalizing factor and conjugate surd. For binomial surds for example $2+\sqrt{3}$ is conjugate of $2-\sqrt{3}$ and it is also rationalizing factor of $2+\sqrt{3}$. ...
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### If the $100$-th derivative of $f$ vanishes on $\Bbb R$, then $f$ is a polynomial.

I have the following statement: If $f^{100}(x) = 0$ for every real number $x$, then $f$ is a polynomial. I couldn't find a counter example so I would like to get some help for prove/disprove. ...
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### Prove the polynomial $P_a=X^5 + a$ is reducible over a field

Let $(K, +, \cdot)$ a finite field so that the polynomial $P=X^2-5$ is irreducible. Prove that: a) $1+1 \ne 0$ b) The polynomial $P_a=X^5 + a$ is reducible $\forall a \in K$ a) ...