# 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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### Roots of polynomial with positive coefficients

My question is very simple. Suppose we have a polynomial defined as follows: $$p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots+a_0$$ where all of the $a_n$'s are all real and positive. Is there something ...
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### Valuation of discriminant

So the discriminant of a polynomial of degree $n$ in the form of determinant of the resultant matrix can be written as $$\det(D)\det(A-BD^{-1}C)$$ where $A, B, C, D$ are block matrices of the ...
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### Construction of a 8-degree polynomial with 16 real numbers

(Vietnam TST 2016/6) Given $16$ distinct real numbers $\alpha_1,\alpha_2,\ldots,\alpha_{16}$. For each polynomial $P$, denote $$V(P)=P(\alpha_1)+P(\alpha_2)+\cdots+P(\alpha_{16}).$$Prove that there ...
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### Are constants a special case of coefficients?

What I hope to understand better, is the relation between constants and coefficients. Consider the following polynomial: $$3x^2+2x+5$$ What are the coefficients in the expression? Obviously, 3 and 2 ...
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### Two questions on the Gaussian integers [duplicate]

I have two questions on the Gaussian integers. Is any element in $\mathbb{Z}[i]$ the root of a monic polynomial with coefficients in $\mathbb{Z}$? Conversely, does any element in $\mathbb{Q}(i)$ ...
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### If $A[X] \cong B[X]$ as rings, are the degrees of irreducible polynomials the same in $A$ and in $B$?

First, I ask my question and then I add some explanations: Suppose that $A$ and $B$ are two commutative rings such that $A[X] \cong B[X]$ as rings. Denote by $D_A$ the set of all positive integers ...
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### Is there a “concatenation operator” for polynomials?

Wikipedia says that the concatenation operator $\|$ concatenates digits of two numbers: ... the concatenation of 69 and 420 is 69420. Is there a similar concatenation operator (or the same?) for ...
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### Finding the roots of an octic

I'm trying to solve a problem, but it involves finding the exact roots of the octic polynomial $$x^8+4x^7-10x^6-54x^5+9x^4+226x^3+125x^2-301x-269$$ How can I find the roots of an octic? Wolfram ...
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### Does there always exist an irreducible polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$? [duplicate]

Let $p$ be a prime and let $d$ be a positive integer. Does there always exist an irreducible (i.e. unfactorable) polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$?
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### The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. [on hold]

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. Find the value of $$\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}$$
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### Find the values of k for which the equation $x^3-9x^2 +24x +k =0$ may have multiple roots and solve the equation in each case.

This question is from Theory of Equations. Help please.
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In my math algebra class my teacher says if $$(1+n)^3=A+B(n)+C(n)(n-1)+D(n)(n-1)(n-2)$$ And solve to find A,B,C,D.I know how to solve it. But I won't understand what it really mean and why he says ...
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### Zeros of a polynomial. [on hold]

If $F(i, x)$ is a polynomial where $i$ is a parameter and $\rho$ is the largest root of $F(0,x)$ and $F(i+1,x)\ge F(i, x)$, Prove that as $i$ increases $\rho$ will increase. I don't understand ...
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### Finding the $\gcd$ of polynomials in $\Bbb R[x]$

Let $f(x)=6x^3-10x^2-6x+10$ and $g(x)=3x^2-14x+15$ in $\Bbb R[x]$. I want to find the $\gcd$ of these two polynomials. I am not really sure how to do this in general, but my approach was as follows: ...
### How to factorise $(x-1)^2 - (x-5)^2$
My attempt: $a = (x-1)$ $c = (x-5)$ $a^2 - c^2$ which is equal to: $$((x-1) - (x-5))((x-1)+(x-5))$$ But the correct answer is : $8(x-3)$ Can you explain, please?