Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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Self-contained formal polynomial reference

In the forward to the third edition of his Undergraduate Algebra, Lang mentions: A new section in Chapter IV gives a complete account of the Mason-Stothers theorem about polynomials, with Noah ...
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Polynomial transformation of the roots of another irreducible polynomial.

Suppose I have some monic irreducible polynomial $g(x)$ in $\mathbb{Z}[x]$ with distinct roots $r_1,r_2,\dots,r_n$. Suppose $f(x)$ is some other polynomial, not necessarily irreducible. Is there ...
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Find prize per unit that will maximize profit at a given x-value

Struggling while reviewing my old math books. The problem has a prize-function and wants to know how the prize-per-unit should be chosen to maximize the profit at x=160. The prize-function is: $$ ...
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Show that $V = {f ∈ R[t] | f(−a) = −f(a) ∀a ∈ R}$ is a vector space by scalar multiplication and addition [on hold]

How would i show this? would i do there exists and $f, g ∈ R[t]$ and continue? Also the mark scheme added that V consists of polynomials having uniquely terms of an odd degree. How did they find this ...
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Does $f(n,z)$ have $2^n$ distinct fixpoints $z$ for all $n$?

Let $f(z)$ be a given degree $2$ polynomial. Let $n$ be a positive integer. Let $f(1,z) = f(z)$ and $f(n,z)= f(1,f(n-1,z))$. How to decide if $f(n,z)$ has $2^n$ distinct fixpoints $z$ for all $n$ ?
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Achieving a polynomial that maps from $GF(p^q)$ to {0,1} with the same probability

I am using an arithmetic circuit, which can compute polynomials over the field $GF(p^q)$. I need a polynomial that maps any element from the field to an element from $\{0,1\}$, I need that the range ...
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142 views

Factoring the following polynomials

Factorize the following polynomial: $t^3 -9t +8$ $t^6 -91t^2 +90$
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Constructing a polynomial bump function

Proposition: Suppose $f$ is continuous and $\int_a^bf(x)x^ndx = 0$ for all $n$. Then $f$ is zero on $[a,b]$. This can be proven by uniformly approximating $f$ with polynomials via the Weierstrass ...
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A complex polynomial in $z$ and $\bar z$ contains no terms with $\bar z$ if and only if its $\bar z$-derivative is zero

I am struggling with this exercise: Let $p(x,\bar{z})=\sum a_{lm}z^l\cdot\bar{z}^m$ be a polynomial in $z$ and $\bar{z}$ (so only finitely many $a_{lm}$ are non-zero). Show that p contains no ...
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Solving a Bernstein Polynomial for 3D space (trivariate)

I'm writing a piece of software and need to deform points in 3D space by a set of control points. After some searching I found this paper on how to do it. The summary is 'The deformation function is ...
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Natural cubic spline interpolation - check and suggest better way

I was given the following interpolation nodes: $(0,10),(\frac{1}{2},8),(1,5),(2,2),(3,1)$ and I was asked to find the natural cubic spline interpolation between every 2 points. I want to show you ...
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1answer
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Square-free factorization of polynomials over finite fields

For any $f\in\mathbb{F}_q[X]$, I want to derive an algorithm which computes a factorization $$f=\prod_{i=1}^kf_i^i\tag{1}$$ with square-free polynomials $f_i$. My Ideas: If $f'=0$, we're done ...
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How to prove that all primitive polynomials are irreducible

Let $F$ be a finite field, and $F[X]$ set of all polynomials in $F$, how to prove that: why all primitive polynomials $\;$ $f \in F[X]$ $\;$ must be an irreducible. Note: Polynomial primitive is an ...
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Why do we interpolate - no guarantee of success

this is somewhat of a general question about interpolation, I don't fully understand how can we be confident that our approximation is good, even if we know a lot of points. An example would be: ...
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How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$? The explicit question out of the ...
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Construction of a polynomial of degree 4 with some conditions

Exercise Let $P(x)$ be a polynomial of degree $4$, the question is : Find this $P$ such that : The coefficient oh highest degree is $1$ P is divisible by $x^2+x+1$ The rest of the division of $P$ ...
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Proving degree $n$ have at last $n$ roots in $F_q[X]$

How to prove that in $F_{q}[X]$ of degree $n$, have $n$ roots?
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66 views

Equation $3x^4 + 2x^3 + 9x^2 + 4x + 6 = 0$

Solve the equation $$3x^4 + 2x^3 + 9x^2 + 4x + 6 = 0$$ Having a complex root of modulus $1$. To get the solution, I tried to take a complex root $\sqrt{\frac{1}{2}} + i \sqrt{\frac{1}{2}}$ but ...
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Characterizing kernel of ring morphism

Let $K$ be a field and define a ring morphism $\psi: K[x_1,x_2, \dots , x_n, y_1, y_2, \dots , y_n] \rightarrow K(x_1,x_2, \dots , x_n)$ by $\psi(x_i) =x_i$ and $\psi(y_i) =\frac{1}{x_i}$. I think ...
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Upper bound on the magnitude of the roots of a complex polynomial

Problem: Let $z_0$ be a root of the complex polynomial $z^n + a_{n-1}z^{n-1} + ... + a_0 $ $ (a_k \in \mathbb{C})$. Prove that $|z_0| \le \zeta$, where $\zeta$ is the only positive root of $z^n - ...
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If $p(z)$ is an injective polynomial $\Longrightarrow$ $p(z)=az+b$ [on hold]

If $p(z)$ is an injective polynomial, how to prove that $p(z)=az+b$ with $a\neq 0$. $p(z)\in\mathbb{C}[z]$. Any hint would be appreciated.
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If $p(z)$ is a monic polynomial then $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$

I need some help with this problem: If $p(z)$ is a monic polynomial of degree $n$ then there is a $b\in\mathbb{C}$ such that $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$ where $z_1,z_2,\dots,z_n$ are simple ...
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Division by factorized polynomials in Macaulay2

I have this problem dividing by factorized polynomials, for example (x_1^4-x_2^4)//(factor(x_1^2-x_2^2)) does not work because the numerator is of "class R" (R is the ring kk[x_1..x_n]) and the ...
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Proper Field extensions

Given a field $F$, is there a proper field extension $K$ such that any root in $K$ of a polynomial in $F[X]$ is in $F$? Note: I am not looking for the algebraic closure of $F$. One candidate is the ...
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A problem on polynomial completely

$P(x)=x^3+mx^2+nx+14$ is divisible by $(x+2)$ but leaves a remainder of $-20$ when it is divided by $(x-2)$. Find the values of $m$ and $n$. Hence, factorise the polynomial completely. Now, I get ...
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factorisation over a galois field

I got a question about two examples in my studybook about the factorisation of a galois field. I have included a screenshot of both my examples along with some clarification as it's written in Dutch, ...
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difference between the polynomials

I have a homework assignment that I do not know how to solve. I don't understand how to calculate $f(x)$ in this assignment. $f(t)$ is the difference between the polynomials $2t^3-7t^2-4$ and ...
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existence of a positive root

Consider the polynomial $$ P(\omega)=\omega^8+\phi_7\omega^7+\phi_6\omega^6+\phi_5\omega^5+\phi_4\omega^4+\phi_3\omega^3+\phi_2\omega^2+\phi_1\omega+\phi_0 $$ with real coefficients. Assuming that ...
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Find a generator for an ideal in $\mathbb{Q}[T]$

Let $I$ be the ideal in $\mathbb{Q}[T]$ generated by $L=\{T^{2}-1, T^3-T^2+T-1,T^4-T^3+T-1\}$. Find $f\in\mathbb{Q}[T]$ such as $(f)=f\mathbb{Q}[T]=I$. The book solution proves that $I\subseteq ...
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Polynomial modulus in Quotient Ring

I have a ring $R=\Bbb Z[x]/(x^m+1)$ with $m$ some power of two and a polynomial $g \in R$, which has relatively small coefficients and some other properties that I believe to be irrelevant for this ...
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Algorithms for solving overdetermined, homogeneous linear systems with multivariate polynomial coefficients

I would like to solve overdetermined, homogeneous linear systems of equations with multivariate polynomial coefficients, i.e., $Ap=0$ with $A$ an $m\times n$ matrix, $m\gg n$, and $a_{i,j} \in ...
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Find isomorphism between $\mathbb{Q}[T]/(T^2+3)$ and $\mathbb{Q}[T]/(T^2+T+1)$

The books states that the isomorphsim is $g(T)=2T+1$ and the identity when restricted to $\mathbb{Q}$. I would like some help to understand what the process is to find $g$.
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Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7)

Solving a system of polynomial equations in three variables (x^2-yz=18, y^2-zx=8, z^2-xy=-7 I've tried rearranging each equation to isolate for one variable ex: z^2-xy=-7 --> z= x^2-18/y after, I ...
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System of quadratic equations that is symmetric

Solve for $z$: $z^2-3z+1=x, x^2-3x+1=z$ I see that it is symmetric, but not anything else. Hints would be great, but please do not spoil the answer. Thanks!
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1answer
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Use $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial?

I am reading Barbeau's Polynomials and I found the following problem: Use the identity $\sin^22t=4\sin^2t(1-\sin^2 t)$ to show that $\sin t$ is not a polynomial. But I really have no idea on how ...
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Every time a real solution.

I have got an interesting exercise. Proof that for all positive integer $a$ and $p(x) = x^2+2013x + 1$, $\underbrace{p(p(\dots p}_{a \ \ \text{times}}(x)\dots )) = 0$ has got at least 1 real solution ...
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1answer
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Silly number theory questions I can't prove.

I know if $gcd(r,s)=1$ then $1=as+bs$ for some intgers $a,b$. Here's what I want to know: which numbers can be written as $as+bs$, if I am restricted to $a,b \in \mathbb{N}$? To be more specific, I ...
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Find $g(x)$ if $(x^2+a^2)(x^2 + b^2)(x^2 + c^2) = (f(x))^2 + (g(x))^2$ and $f(x)$ is a degree three polynomial [closed]

If $$(x^2+a^2)(x^2 + b^2)(x^2 + c^2) = (f(x))^2 + (g(x))^2$$ where $f(x)$ is a degree three polynomial, find $g(x)$.
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Can someone help me to prove this theorem from Axler's *Linear Algebra Done Right*?

If $p\in P(\Bbb{R})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form ...
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Irreducible polynomial

Does there exist an irreducible polynomial over a field K with two roots $a,b$ and $k\in K$ such that $a=b+k$ ? This can't happen if K is of characteristic $0$ , but can it happen if K is of ...
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Different forms of a quadrature

I am solving the following problem: Find the quadrature of the following form: $Q(f) = Af(−1) + Bf(0) + > Cf(1)$, which has the highest degree and interpolates the integral: $\int_{-3}^{3} ...
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the numer of monic irreducible polynomials of degree $3$ in $\mathbb{F}_q$

I want to know how hany monic irreducible polynomials of degree $3$ there are in a field $\mathbb{F}_q$. The whole number of monic polynomials of degree three is $q^3$. Now I want to find out how ...
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1answer
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Proof the Existence and Uniqueness of Factorization Form of Polynomial with Complex Coefficient

If $p\in P(\Bbb{C})$ is a nonconstant polynomial, then $p$ has a unique factorization (except for the order of the factors) of the form $$p(z)=c(z-\lambda_1)....(z-\lambda_m)$$ where ...
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Which polynomials fix the unit circle?

Find all polynomials $P(x)$ with real coefficients such that for every $x,y\in \mathbb{R}$ satisfying $x^2+y^2=1$ we have $$P(x)^2+P(y)^2=1$$
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Chinese remainder theorem for polynomial evaluation

Let $R$ be a euclidean domain, $m_0,\ldots ,m_{k-1}\in R$ be pairwise coprime and $m:=m_0\cdots m_{k-1}$. The Chinese remainder theorem states: $$\varphi:R\to R/(m_0)\times\cdots \times ...
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Derivatives of Lagrange polynomials

It seems there is some relationship between Lagrange polynomial and Legendre polynomial. That is Lagrange polynomial can be expressed as a function of Legendre polynomial. If so, I could use this ...
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How to multiply the binomials $(2x^3 - x)\left(\sqrt{x} + \frac{2}{x}\right)$

I am sorry if the numbers are not formatted, I have searched but found nothing on how. I am trying to multiply $$(2x^3 - x)\left(\sqrt{x} + \frac {2}{x}\right)$$ together and I arrive at a different ...
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If a Sequence of Polynomials Converge to Another Polynomial Then the Roots Also Converge.

Proposition 5.2.1 in Artin states that: THEOREM. Let $p_k(t)\in \mathbf C[t]$ be a sequence of monic polynomials of degree $\leq n$, and let $p(t)\in \mathbf C[t]$ be another monic polynomial ...
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254 views

How can I show why this equation has no complex roots?

I've been asked to show why an equation has no complex roots but i'm at a complete loss. The equation is $F_{n+2}=F_n$ Where $F_n=(x-1)(x-2)...(x-n)$ and n is a positive integer. I'd really ...
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Show that if $\mathrm{Tr}(y)=0$ then there exists a $x$ such that $x^p-x=y$.

We have the Trace map defined by: $$ \mathrm{Tr}\colon \mathbb{F}_q\rightarrow\mathbb{F}_q\colon x\mapsto x+x^p+x^{p^2}+\cdots+x^{p^{n-1}}, $$ where $q=p^n$. Now I have to prove that if ...