Questions tagged [polynomials]

For both basic and advanced questions on polynomials in any number of variables, including, but not limited to solving for roots, factoring, and checking for irreducibility.

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Given matrix associated to linear transformation find the basis it corresponds to

I know how to find the matrix associated to a linear transformation with respect to two given basis, but how can one find the basis given the matrix? Let's say we're given a linear transformation $T : ...
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1 answer
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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 $$ 12\sqrt x = x^2 - 7x + 12 $$ $$ 12 \sqrt x = (x-4)(x-3) $$ ...
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Examples of finite group representations on polynomials

Studying the representations of Lie groups on polynomials has been very valuable to me. For example, the natural action of $SO(3)$ on real homogeneous polynomials of 3 variables, or the action of $SU(...
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If a sequence is generated by a $\mathbb{Q}$-polynomial passed mod $p$, can we find an appropriate polynomial over an extension of $\mathbb{F}_{p}$?

If we have a polynomial that takes integer values for integer inputs, we can take its outputs at integer inputs and pass them $\text{mod }p$. However, my understanding is that the coefficients of the ...
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Determining the signature of a Matrix based on the characteristic polynomial.

Let $A$ be a hermitian matrix with the characteristic polynomial $p_A=a_0+a_1x+...+a_nx^n.$ Furthermore let $p$ be the number of sign changes in the sequence $\{a_0,a_1,...a_{n-1},1\}$ and $q$ be the ...
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Compare weak and finest locally convex topology on $\mathbb{R} [x]$

Let $V = \mathbb{R} [x] \cong \bigoplus_{\mathbb{N}} \mathbb{R}$ be the vector space of univariate polynomials, or the space of real sequences that have all but finitely many elements equal to zero. ...
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1 answer
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In what sense is $\Bbb R(x)$ an "instantiation" of the hyperreals?

I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the ...
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$\mathrm{Res}(f, g, h)$ as a product of three resultants?

I'm currently working with resultants, which I define as follows: let $k$ be a field, $V$ be a 2-dimensional vector space over $k$, and let $S^dV^*$ be the $d$-th symmetric power of dual space, i.e. ...
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2 votes
0 answers
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Generalizing Ramanujan cubic denesting formula to higher powers

We have the following theorems for denesting radicals of degree 2 and 3 : Denesting theorem for degree 2 : If $\alpha, \beta$ are the roots of the equation, \begin{equation} x^2-ax+b = 0 \end{equation}...
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Understanding the difference between a divisor and a factor

Consider an arbitrary polynomial of degree four $$ P(x)=ax^4+bx^3+cx^2+dx+e,$$ where $a,b,c,d,e\in\mathbb{R}$. If $P(x)$ is divisible by a quadratic, say $(x-1)(x+9)$, are $x-1$ and $x+9$ factors of $...
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On the number of roots of $p(z,\bar z)$

Let $p$ be a polynomial with complex (or even real) coefficients in the variables $z$, $\bar z$, where $\bar z$ is the conjugate of $z$. What can we say on the number of complex roots of $p$? Clearly $...
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Polynomial interpolation yielding integer coefficients

Let $y_1,\dots,y_n$ be integers. Conjecture:$\;$There exists $f\in\mathbb{Z}[x]$ such that $f(i)=y_i$ for $i=1,\dots,n\;$if and only if $(i-j){\,\mid\,}(y_i-y_j)$ for all $i,j$ with $1\le j < i\le ...
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Showing that if the parametrizing values $a_1,\dots,a_n$ of a Vandermonde matrix are unique then it has a non-zero determinant with the use of the FTA

My whole title would be: Showing that if the parametrizing values $a_1,\dots,a_n$ of a Vandermonde matrix are unique then it has a non-zero determinant, with the use of the fundamental theorem of ...
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Fraction field and Monic Polynomial

Here is the question. $R$ is a UFD and $F$=Frac$R$, $E=F(α)$ is an algebraic extension of $F$. Prove of disprove that, if the minimal polynomial of $α$ over $F$ belongs to $R[x]$, then $α$ is a root ...
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Finding the kernel of a given map

Let $\mathbb K$ be an algebraic closed field and let $f,g\in\mathbb K[X,Y]$ be two polynomials so that $V_{\mathbb K}(f)$ and $V_{\mathbb K}(g)$ don't have common irreductible components and $(0,0)\in ...
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Finding Polynomials with negative coefficients

There is a well known mathemagic trick where a volunteer from the audience comes up with a single variable polynomial with positive integer coefficients and no like-terms. The mathemagician then can ...
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Finding a polynomial satisfying $f(a)=A$, $f'(a) = 0$, $f(c)=C$, $f'(c)=0$, $f'(b)=m$

I'm looking for a curve (polynomial?) which satisfies the following constraints: $$ f(a) = A, f'(a) = 0 $$ $$ f(c) = C, f'(c) = 0 $$ $$ f'(b) = m $$ $$ A < C , a < b < c, m > 0 $$ ...
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Can $p\big(\frac{z+z^{-1}}{2}, \frac{z-z^{-1}}{2i}\big)$ vanish for every $z$, for some polynomial $p$ in two complex variables?

Is there some non-vanishing polynomial $p$ in two complex variables, such that $p\big(\frac{z+z^{-1}}{2}, \frac{z-z^{-1}}{2i}\big) = 0$ for every non-zero complex number $z$? I'm only interested in ...
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9 votes
1 answer
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Easier way to solve equation systems of $a+b+c+\cdots{}= 1$, $a^2 + b^2 + c^2+\cdots{}=2$ and so on without having to crunch massive expressions

I study at below college level. I have been trying to solve certain systems of equations involving $n$ equations of $n$ unknowns. For example, for $2$ unknowns, the problem is \begin{align} a^{\...
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1 vote
1 answer
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Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010.$

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010.$ I know that if the coefficients and roots are integers then for every $r \in \...
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0 answers
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Reasoning behind the rejection of possibility of infinite degree polynomial to show that cos x cannot be written as a polynomial in sinx

https://math.stackexchange.com/a/3954/961436 . In the given solution for showing cosx cannot be written in terms of a polynomial in sinx it is said there that : " "By squaring we see that ...
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1 answer
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Symbol and principal symbol of differential operator

The symbol of a differential operator is defined as $L(x,p):=\sum_{|\alpha| \leq m}a_\alpha(x)p^\alpha$ and the principal symbol $L^p(x,p)$ is defined similar but with $|a|=m$. What would be the ...
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1 vote
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What is nature of root of the polynomial $x^5 -10x + 20?$

Problem The polynomial $x^5 -10x + 20$ has a. both positive and negative real roots b. only positive real roots c. only negative real roots d. at least two complex roots My Approach Tried to solve ...
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1 vote
2 answers
28 views

Regarding the solution of finding the remainder of $g(x^{12})$ divided by $g(x)$

Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12})$ is divided by the polynomial $g(x)$? I'm reading the solution for this and I don't understand how can ...
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Proving that cubic polynomial has no real roots in some range (0, a)

Suppose we have the univariate cubic polynomial $$f(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0 , \quad x \in (0, z_1), $$ where $a_j = g_j(z_1,z_2)$ are rational functions of some real positive constants $...
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1 answer
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Finding functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0} $

Undergraduates at my university showed me this problem, which I found intriguing and now want to see the solution of: Find all functions $f, g, h : \mathbb R_{> 0} \to \mathbb R_{> 0} $ such ...
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1 answer
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How to find modular roots of $x^{22}-2x^{11}-x+2$ (to show it has more than $22$ solutions by CRT).

Consider a polynomial $P$ defined by $P(x)=x^{22}-2x^{11}-x+2,$ how to show that there exists an integer $n\geq1$ such that the equation $P(x)\equiv0$ modulo has more than $22$ solutions modulo $n?$ *...
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Find the number of positive integers $n,$ $1 \le n \le 100,$ for which $x^{2n} + 1 + (x + 1)^{2n}$ is divisible by $x^2 + x + 1.$ [duplicate]

Find the number of positive integers $n,$ $1 \le n \le 100,$ for which $x^{2n} + 1 + (x + 1)^{2n}$ is divisible by $x^2 + x + 1.$ I cannot figure this one out. If $x^{2n} + 1 + (x + 1)^{2n}$ is ...
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1 vote
2 answers
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How can other unknowns be calculated in polynomial division given there are no remainders?

I’ve being practicing polynomial long division for the last week and have built some competence/confidence around the algorithm for performing the operation, but this is stumping me: Given P($x$) = $(...
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-3 votes
1 answer
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Can roots of quintic polynomials be a solution of radicals? [closed]

Assuming integer co-efficients, does there exist a solution to a quintic polynomial that is a solution of radicals? I understand that there is no general formula to solve any quintic, but that doesn't ...
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0 answers
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The polynomial $p(x)$ satisfies $p(1) = 210$ and $(x + 10) p(2x) = 8(x - 4) p(x + 6)$ for all real numbers $x.$ Find $p(10).$

The polynomial $p(x)$ satisfies $p(1) = 210$ and $$(x + 10) p(2x) = 8(x - 4) p(x + 6)$$for all real numbers $x.$ Find $p(10).$ Noting that for $x=4$ we get that $p(8)=0$, for $x=2$ we have that $p(4)=...
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4 votes
1 answer
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$P(x)$ is odd-deg polynomial with real coefficients. Show using Induction that $P(P(x))=0$ has atleast as many distinct real roots as the $P(x)=0$.

PROBLEM Let $P(x)$ be an odd degree polynomial in $x$ with real coefficients. Show using Induction that the equation $P(P(x))=0$ has at least as many distinct real roots as the equation $P(x)=0$. MY ...
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2 votes
4 answers
128 views

If $f(x)$ is irreducible, is $f(x^k)$ irreducible?

Let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $\ge 2$. Is it true that $f(x^k)$ is irreducible for $k\ge 2$? If not true, under what hypothesis, we can gurantee positive answer? ...
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Are there other expressions similar to polynomials in their ease of calculation and customizability?

I have a bunch of 2D squiggles and I need to get their y value based on an input x value. All of them can be calculated through some kind of y = f(x), as in each y value has one and only one ...
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2 votes
1 answer
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prove that the polynomial $(y-1/2)^2 -1/8$ minimizes its variation away from $0$ on $[0,1]$

Let $P(y)$ be a monic quadratic polynomial. Define the variation of P away from $0$ on $[0,1]$ to be $\max_{y\in [0,1]} |P(y)|$. Prove that $P(y) = (y-1/2)^2 -1/8$ has the minimum variation away from $...
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1 vote
0 answers
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Equality of two free modules of the same finite rank under strong hypothesis.

So basically the question is the one of the title of the post, but let me show you the context: Let $\mathbb K$ be an algebraic closed field and let $f\in \mathbb K[X,Y]$ be a polynomial such that is ...
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-1 votes
0 answers
14 views

Cubic Equation Roots for Compensator [closed]

I am currently stuck at designing a Compensator for a control system. with the closed loop equation : x^3 + 3x^2 + 2x + 1.06 how do I solve this cubic equation to get the roots with imaginary parts?
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1 vote
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(extremum)Let $f: \mathbb R \to \mathbb R$ be a polynomial function ...

Let $f: \mathbb R \to \mathbb R$ be a polynomial function $f = a_0 + a_1x + … + a_n x^n$. Let $a_1 = a_2 = … = a_k = 0$, ($k$ less than n) and $a_{k+1} \ne 0$. The function $f$ has an extremum at the ...
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Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1$, $f(2) + f(3) = 125,$ and for all $x$, $f(x)f(2x^2) = f(2x^3 +x)$. Find $f(5)$

Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1$, $f(2) + f(3) = 125,$ and for all $x$, $f(x)f(2x^2) = f(2x^3 +x)$. Find $f(5)$. If $r$ is a root of $f$, then $f(r)f(2r^2)=f(2r^...
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0 votes
2 answers
61 views

An equivalence condition for $\mathcal{I(V}(I)) = \sqrt{I}$ over real fields

Question: Does $\mathcal{I}_{\mathbb{R}}\mathcal{V}_{\mathbb{R}}(I) = \sqrt{I}$ imply that $\overline{\mathcal{V}_{\mathbb{R}}(I)} = \mathcal{V}_{\mathbb{C}}(I)?$, where $I$ is an ideal of $\mathbb{R}[...
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0 answers
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Condition for which the real roots of the polynomial is none

https://math.stackexchange.com/a/1302643/1021792 in this solution given it was being said as coefficients are all positive hence the equation $x^6 + 4x^5 + 5x^4 + 4x^3 +2x +1 = 0$ has no real number ...
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2 answers
69 views

Show for $f(x)=x^4 + 2 x^3 + 3 x^2 + 3x + 1$ that if (real) $x$ satisfies $f(f(x)) = x$ then in fact $f(x) = x$

Define $$f(x)=x^4 + 2 x^3 + 3 x^2 + 3x + 1.$$ Show that if $f(f(x)) = x$ then $f(x) = x$. We can write $f(x)=x^4+2x^3+3x^2+x+1$ and $x=f(x)^4+2f(x)^3+3f(x)^2+f(x)+1$, and subtracting the two ...
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1 vote
1 answer
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Prove the zeroes of a polynomial are all real and distinct

For a polynomial $P(x) = (x-x_1)(x-x_2)\cdots (x-x_n)$ with distinct real zeroes, $x_1 < x_2<\cdots < x_n$, prove or disprove that all zeroes of $f(x) := P'(x) - kP(x)$ are real and that for ...
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-5 votes
0 answers
30 views

What is the value of "b" if -b/2a > 0 [closed]

I got this doubt when i was going through my textbook. Each question had different sign for "b".
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3 votes
3 answers
84 views

When is a multivariable polynomial the zero polynomial

It is known that if a single variable polynomial $p$ with degree at most $n$ has at least $n+1$ zeroes then it must be the $0$ polynomial. Is there an easy to use variant of that for multivariate ...
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zeros of the zero polynomial

According to Theorem 4.12 of Linear Algebra Done Right by Sheldon Axler, for every polynomial p $\in \mathcal{P}(\mathbb{F})$ with degree $m\geq0$, the polynomial p has at most m distinct zeros in $\...
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The name for a simple polynomial division fact

Let $A, B, C \in \mathbb{R}_d$ be non-zero at most degree $d$ polynomial with real coefficient. And denote $\lfloor{\frac{A}{B}}\rfloor$ be the quotient of polynomial division. Does this hold $\lfloor{...
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1 vote
2 answers
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Method checking to show the given inequality to be true

Given quadratic polynomial $f(x)$ satisfies $\lvert ax^2 + bx + c \rvert \leq \lvert x \rvert$ for all $x \in [-1,1]$. Show that $\lvert a \rvert + \lvert b \rvert \leq 1$. My approach was graphical ,...
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1 vote
1 answer
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Showing that constant term of the polynomial satisfy the below inequality

If a polnomial $R(x)$ of ninth degree satisfies $|a_9x^9 + a_8x^8+..a_0|$ $\leq$ $1$ $\forall$ x $\in$ $[-1,1]$ - {${0}$} , then show that $a_0$ satisfies $|a_0| \leq 1$. And does equality every ...
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2 votes
1 answer
38 views

Why do we take the degree of numerator 1 degree less than the denominator?

In expanding rational polynomials, I saw that if the degree of the denominator is 2, then the numerator is $Bx+C$. if the degree is 3, then it is $Bx^{2}+Cx+D$. I felt that it might be because we want ...
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