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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Methods for verifying correct factorisation of polynomials

In an attempt to factor using a GCF, Mia wrote $8x^2 + 4x = 4x(2x – 0)$, which is not correct. a. Explain how Mia could check her work. b. What error did Mia make? She didn't factor using the GFC ...
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Bairstow method improvements

I was reading about Bairstow method for polynomial root finding and I find very compelling that it uses just real numbers, as I'm interested in real roots of real polynomials only. However, couple of ...
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How do I use the bowtie method to multiply $(2x-27)(-x+15)$?

The bowtie method seems like an easy concept to have down, but how is it used to multiply binomials such as $(2x-27)(-x+15)$? Calculating the answer is not the problem, because I can get ...
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A question in matrix polynomial [duplicate]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
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How prove : polynomial $P(x)=W^{\prime \prime}(x) +(W^\prime(x))^2$ have a real root. [closed]

Let $W(x)$ be a polynomial of degree> 2 having at least three different real roots. How prove : polynomial $P(x)=W^{\prime \prime}(x) +(W^\prime(x))^2$ have a real root? Whether the assumption of ...
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Factor Polynomial using GCF method

I am totally lost as to how factorize this polynomial using GCF: I am able to solve others at least a couple of them but not sure of this one? Is it that all polynomial can be factorized using GCF ...
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1answer
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Mason's Theorem Proof. From Algebra (S. Lang)

Mason's Theorem Proof. I have a question about the last step in the proof. I will write it. Mason's Theorem states that if $a, b$ and $c$ are relatively prime polynomials such that $a + b = c$ ...
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Lipschitz-like behaviour of quartic polynomials

I have observed the following phenomenon: Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to ...
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Solving a system of five polynomials

I am trying to solve the following system of equations for tuple $\left(a,b,c,d,t\right) \in \mathbb{R}^{4} \times [0,1]$, with parameter $\ell\in\mathbb{R}$. $$ \begin{eqnarray} a\frac{t^{2}}{2} - ...
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How to divide certain polynomials?

Can somebody help me with this question? $$\frac{15p^3+16p^2+46}{3p+5}$$ For some reason I can't wrap my head around the process used to divide polynomials, I can do long division but every time ...
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1answer
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Creating Polynomial Function with Surface Area of Cylinder

I've spent a few hours at this question but can't seem to get the right answer. I was hoping someone here can lead me in the right direction. The question: A storage tank is to be constructed ...
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Literature on Quadratic Residues in Polynomial Quotient ring

I have a literature-question. Given a field $\mathbb{F}$, finite or infinite, and an element $f$ in the polynomial ring $\mathbb{F}[t]$. I am searching for results of any kind about quadratic residues ...
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How many such polynomial exist?

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010$. I got: $$P(x) = ax^2 + bx + c \implies P(0) = c = 2010$$ Let $P(r_1, r_2) ...
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About sparse polynomial squares

Given $p\in\mathbb{Q}[x]$, we define the weigth of $p$ as: $$ W(p) = \#\{n\in\mathbb{N}: [x^n]\,p(x)\neq 0\} $$ i.e. as the number of non-zero terms. By playing a bit with the Taylor series of ...
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Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero coefficients than $f(x)$?

I saw this problem on a problem set and I have absolutely no idea how to proceed in a feasible way. Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero ...
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Generalization of the derivative to polynomial rings

It is easy to see why the derivative plays an important role in real and complex analysis from the geometric viewpoint. However, one can extend the definition of a derivative to polynomial rings such ...
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Proving Things About Rings Using Things About Vector Spaces

All rings below are assumed to be commutative and having an identity. $\newcommand{\bw}{\bigwedge}\newcommand{\R}{\mathbf R}\newcommand{\mc}{\mathcal}$ Consider the following problem: Problem 1. ...
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2answers
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Can a non-rational polynomial be rational at all integers?

Is there a polynomial $f \in \mathbb{R}[X]$ such that for every $x \in \mathbb Z,\>\> f(x)$ is rational but at least one of the coefficients of $f$ is irrational?
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1answer
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How do I properly represent this question? (binomial polynomials)

The total area of a picture and its frame can be represented by (l + 2f)(w + 2f), where l and w represent the length and width of the picture, respectively, and f represents the thickness of the ...
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Help me solve this olympiad challenge?

Given: $$p(x) = x^4 - 5773x^3 - 46464x^2 - 5773x + 46$$ What is the sum of all arctan of all the roots of $p(x)$?
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Irreducible polynomials with one large coefficient

Is it true that for every monic polynomial $p(x) \in \mathbb{Z}[x]$, $p(0)\neq 0$, of degree $n>0$ there exists a real number $M>0$ such that for every $|m|>M$ and for every $k$ odd integer ...
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Prove that $T_n$ satisfy $ \sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} = \begin{cases} 0 &: i\ne j \\ l\neq 0 &: i=j \end{cases} \,\! $

The Chebyshev polynomials of the first kind satisfy the recurrence relation $$ \begin{cases} T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\ T_{0}(x)=1, \ \ T_{1}(x)=x \\ \end{cases} $$ The ...
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Why are the coefficients always equal?

Take the equation $ax^{2} + bx + c = 3x^{2} + 4x + 53$. Why is it always true that $a = 3, b = 4$ and $c = 53$? I've seen many examples like this where the coefficients are equated, and was just ...
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Is it possible to find solutions to polynomials purely by calculus and without iteration?

I know this may sound peculiar, but I was wondering if any mathematicians have found a method to finding roots purely through calculus without iteration. I can't imagine that such a method exists for ...
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2answers
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Locating the roots of a cubic polynomial.

Given a cubic polynomial $f(x) = ax^{3} + bx^{2} + cx +d$ with arbitrary real coefficients and $a\neq 0$. Is there an easy test to determine when all the real roots of $f$ are negative? The ...
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1answer
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Approximating Trig Functions with Polynomials

I was thinking about the graphs of different trig functions and noticed that most of them are of a similar shape to some different types of polynomials. For example: Higher degree polynomials create ...
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Composition factoring into nonnegative integer polynomials

Consider the integer polynomials with nonnegative coefficients, such as: $$ 1 + 2x +2x^2 $$ $$ 3 + 3x^4 + 11x^{10}$$ $$ 13 + 7x + x^2 $$ I asm interested in knowing "what is a composition ...
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Polynomials: Linking equations via their roots

The equation $ax^3+bx^2+cx+d=0$ has solutions $1,p$ and $q$. The equation $x^3+sx^2+tx+r=0$ has solutions $1, 1/p$ and $1/q$. Show that $r=a/d$ , and find an expression for $s$ in terms of $c$ and $d$ ...
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Show that $\mathbb{F}[x^2,y^2,xy]$ is not polynomial

$\mathbb{F}[x^2,y^2,xy]$ is the polynomials in two variables whose terms all have even degrees. Of course, this generating set $x^2,y^2,xy$ is not algebraically independent, but I need to show that no ...
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Guessing one root of a cubic equation for Hit and Trial

Suppose I have a cubic equation as $$15x^3-4x^2-25x+14=0$$ By Hit and Trial method I know that one of the roots is $x=1$. And hence I can solve the cubic equation wit ease as it will take the form ...
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1answer
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how to choose coefficients for a polynomial which gives a pair of complex conjugates at roots?

I recently found WeBWork and was trying to write up some problems and wanted to make a problem such that the factors of the polynomial would have a pair of complex conjugates. I understand that $2$ ...
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Understanding how Prime Polynomials are applied to LFSRs?

In doing some research on LFSRs I understand that a primitive polynomial can determine what taps to be used to create an LFSR that has as many bits as the degree of the polynomial that will cycle ...
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How to make this equation to its solutions were $\frac{a}{b+c-a} ; \frac{b}{c+a-b} ; \frac{c}{a+b-c}$?

Let $a,b,c$ be solutions of $x^3+px^2+qx+r=0$. How can we transform this equation so its solutions are $\frac{a}{b+c-a} ; \frac{b}{c+a-b} ; \frac{c}{a+b-c}$? I know $a+b+c=-p$, $abc=r$, $ab+bc+ca=q$. ...
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Polynomial division remainder

Given the polynomials $f,g,h \in \mathbb{R}\left[X\right]$ with $$f=(x-1)^n-x^n+1$$ $$g=x^2-3x+2$$ $$h=x^2-x$$ where $n\ge3$ Find the remainder of dividing the polynomial f to g. Prove that if $n$ is ...
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Turning real roots into curves (for visualisation)

One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$. Thinking about data visualisation, one can portray a set of $N$ observations ...
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Ideal of polynomials vanishing on $\{(x,y): x^2+y^2=1, x \neq 0 \}$

I'm reading the book "Introduction to algebraic geometry" by Hassett, and in Chapter 3, after introducing the concept of the ideal of polynomials vanishing on a set $S$, the author gives some ...
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Cubic equation $X^3+pX+q$ not solvable by radicals if $D=-4p^3 - 27q^2 >0$

How can one prove that the real cubic equation $$P(X)=X^3+pX+q$$ is not solvable by real radicals when $$D=-4p^3 - 27q^2 >0?$$ Which means that there is no sequence of extension: $$\mathbb R=L_0 ...
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$5x^{3}-5(p+1)x^{2}+(71p-1)x-(66p-1)=0$ has three solutions which are natural numbers?

How find all values of parameter $p \in R$ such that equation $~ 5x^{3}-5(p+1)x^{2}+(71p-1)x-(66p-1)=0$ has three solutions which are natural numbers? My try $(x - 1)(5x^2 - 5px + 66p - 1)=0$ and ...
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Dividing a polynomial $p(x)$ by $x-k$, but $x-k$ is set to be $0$?

While studying, I read the following, When a polynomial $p(x)$ is divided by $(x-k)$, if we set $(x-k)$ to be $0$, we get $x=k$ and the remainder as $p(k)$. However, if we divide $p(x)$ by ...
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Finding relations of variables

Suppose that \begin{align*} x&=t+t^{-1}+t^2s+t^{-2}s^{-1}+ts^{-1}+t^{-1}s-6\\ y&=t+t^{-2}+ts+s^{-1}-4\\ z&=t^{-1}+t^2+t^{-1}s^{-1}+s-4 \end{align*} Find a polynomial $P(x, y, z)=0$ ...
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Algorithmic question about algebraic varieties and affinely independence

Let $f \in \mathbb{R}[x_1,\ldots,x_d]$ be a polynomial in $d$ variables. Then we can write $f$ as $$ f = \sum_{i=0}^m f_i x_1^i $$ where $f_i \in \mathbb{R}[x_2, \ldots, x_d]$. We may assume $m \neq ...
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Legendre Polynomials: proofs

Does any one know, how to compute any of those two things? The relationship between Legendre polynomials and Shifted Legendre Polynomials. $\displaystyle\int_{-1}^1P_n^2(x)dx=\dfrac{2}{(2n+1)}$ for ...
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Estimating line paths in vector fields.

Assume I have a vector field sampled in discrete points. For simplicity let us assume it is sampled regularly on a Cartesian grid. I want to estimate flow lines through various points in this vector ...
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Is the cubic formula numerically unstable?

Are there numerical rounding issues in using the cubic formula to find roots of cubic equations? Similarly with the quartic formula? I do know for the quadratic formula to solve $ax^2+bx+c = 0$ that ...
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1answer
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Polynomial inequalities vs rational inequalities

A question from one of the comprehension questions I have is: How would the intervals of the solution set differ between a polynomial inequality and a rational inequality? I have tried to research ...
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29 views

linear combination of polynomial equal to zero

I have a trouble with the following question. Let $p,q \in K[x]$. There are polynomials $a(x),b(X) \in K[x]$ with $\deg(a) < \deg(q)$ and $\deg(b) < \deg(p)$ such that $$a(x)p(x) + b(x)q(x) = ...
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1answer
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Find polynomials $q(x)$ and $r(x)$ such that $f(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or $\deg r(x)<\deg g(x)$

Find polynomials $q(x)$ and $r(x)$ such that $f(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or $\deg r(x)<\deg g(x)$ provided that $f(x)=2x^4+x^2-x+1$ and $g(x)=3x^2+2$ in $\Bbb Z[x]$. The problem I'm ...
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Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $n\ne 4$, is irreducible over $\mathbb Z$

I try to solve this problem. I seems to come close to the end but I can't get the conclusion. Can someone help me complete my proof. Thanks Show that the polynomial $h(x) = ...
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1answer
34 views

Logical proof of remainder theorem

If $p (x)$ is divided by a linear polynomial $(x-a) $then the remainder is $p(a)$. We know,by lemma, that $p(x)=(x-a)q(x)+r(x)$. If we substitute $x$ by $a$ then $r(x)=p(a)$. How can we prove this ...
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1answer
378 views

Working with casus irreducibilis

I read about casus irreducibilis here. As an example of casus irreducibilis, it says we can factor $x^3 - 15x - 4$ to find $4$ as a root and it also has two other real roots. Using Cardano's method we ...