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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2
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26 views

Closed-form solution to polynomials of this special form?

I have two sets of real numbers $\{a_k\}$ and $\{b_k\}$ such that: $$a_n \leq \ldots \leq a_1 < b_1 \leq \ldots \leq b_n$$ Now consider the following polynomial equation: $$\prod_{k=1}^n (x - ...
2
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0answers
55 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
0
votes
3answers
37 views

Help solve rational expression

I need help solving this rational expression. Divide $$\frac{4x^4 + 6x^3 + 3x - 1}{2x^2 + 1}$$ How do you solve this problem? Where do I start?
6
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1answer
56 views

Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $

If $x\in\mathbb{R}$ find the maximum value of $$ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $$ I tried this: Let $$y= \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$$ For maxima ...
0
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3answers
8 views

Setting up word problem for finding length and width

Word Problem: The length of a rectangular sign is $3$ feet longer than the width. If the sign has space for $54$ square feet of advertising, find its length and width. I have not idea where to start. ...
0
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1answer
40 views

Solving the polynominal: $s(t) = -16t^2 + 48t + 160$

The height of a ball is thrown directly upward from an initial height of $160$ ft with an initial velocity of $48$ ft per second is given by the function: $s(t) = -16t^2 + 48t + 160$, where $s(t)$ ...
1
vote
1answer
35 views

How to find the solution of a quadratic equation with complex coefficients?

I know how to find the solution for a quadratic equation with real coefficients. But if the coefficient changes to complex numbers then what is the change in the solution? Want an example of such ...
5
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0answers
249 views

A problem with sign of coefficients of a polynomial expression

Let $f$ be a real coefficient homogeneous polynomial in $n$ undeterminates, such that $f(x_1,\cdots,x_n)>0$ whenever $x_1,...,x_n$ are non-negative real numbers, not all $0$. Then how to show ...
1
vote
1answer
55 views

I need help proving a statement about rational roots

I have no idea where to start...this is the statement: If a polynomial of degree not greater than 5 with rational coefficients has multiple roots, it has also a rational root, except in the case ...
1
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0answers
32 views

Decomposition of polynomials and inequality

This was asked in comment here by @23rd : If $f$ is a polynomial with $\deg f=n\ge2$, then there exist polynomials $g$ and $h$, such that $$f(x)=2xg(x)−h(x)$$ $$\deg g\le n−1, \quad \deg h \le ...
4
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1answer
272 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
2
votes
1answer
37 views

Determine the coefficients of a polynomial knowing its roots

My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice. Given the following polynomial: ...
17
votes
2answers
926 views

Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that:

Let $a_{i} \in\mathbb{R}$ ($i=1,2,\dots,n$), and $f(x)=\sum_{i=0}^{n}a_{i}x^i$ such that if $|x|\leqslant 1$, then $|f(x)|\leqslant 1$. Prove that: $|a_{n}|+|a_{n-1} | \leqslant 2^{n-1}$. ...
0
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1answer
46 views

Polynomial long division to determine the quotient [closed]

Use polynomial long division to determine the quotient when 3x^3 - 5x^2 + 10x +4 is divided by 3x + 1?
4
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2answers
63 views

Need help with a diophantine expression

I'm faced with this problem. Under what conditions is this expression a positive odd integer: $$\frac{2^g(x^2+y^2-z^2)}{x+y-z}$$ where $g,x,y,z$ are nonnegative integers. x and z are odd, and y is ...
0
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0answers
36 views

Construct a polynomial with a certain root.

Suppose we have a polynomial $g(x)=ax^3-bx^2+cx-d\in\mathbb{Z}[x]$ whose zero is $\rho$. How do we construct a polynomial which involves $g(x)$ and with a root $\frac{d}{\rho}$? I was trying to do ...
0
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6answers
68 views

solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
1
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2answers
72 views

Non linear curve

If there are set of points (16.26,1000) (8.814,2000) (6.06,3000) (4.602,4000) in a similar way if you plot these points in matlab you will get a nonlinear curve. After getting that curve, how to find ...
0
votes
0answers
17 views

Sufficient condition for a indefinite integral to be an elementary function

I would like to find a sufficient condition on two polynomials $P(s)$ and $Q(s)$, such that the function $s \mapsto Q(s)e^{P(s)} $ has a primitive integral of the form $s \mapsto R(s)e^{P(s)} $ (with ...
2
votes
1answer
103 views

Algebraic problem - irreducible polynomials

Let $p$ be a prime number and $q=p^n$ with $n \in \mathbb{N}$. We define the polynomial$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
3
votes
2answers
32 views

Build field extension and solve equation

Build quadratic extension of field that contains $5$ elements. And solve $x^2+x+2=0$ in this field. As I understand we need to build $\mathbb{F}_{5^{2}}$. Field $\mathbb{F}_5$ contains ...
4
votes
1answer
30 views

Polynomials vanishing on subsets of $\mathbb{R}^2$

Let $\mathcal{S}\subset\mathbb{R}^2$ such that every point in the real plane is at most at distance $1$ from a point in $\mathcal{S}$. Is it true that if $P\in\mathbb{R}[X,Y]$ is a polynomial that ...
0
votes
1answer
26 views

remainder is not zero using long division method

Find all zeros of $f(x)=128x^3-48x^2+1$ given that one linear factor occurs twice. let $f(x) $ be equaal to 0 $128x^3-48x^2+1=0,$ $16x^2(8x-3)+1=0,$ trying $x=1/4$ $16/16(2-3)+1=0,$ ...
3
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1answer
44 views

What is the Most Efficient Way to Calculate the Internal Rate of Return?

I have built a program that prices financial assets and it does this in part by calculating the IRR. The problem is that it does not run as quickly as I would like it to. I currently use the ...
-1
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2answers
81 views

How prove the statement that a infinity derivable function has a Taylor expansion. [closed]

What is the proof of the following statement (Taylor statement): An infinity derivable function (like $\sin x$ or $\cos x$) has a infinite polynomial series that approaches it.
8
votes
1answer
351 views

IMO 1979 problem

The question is $$\text{If }\, p, \ q\in \mathbb{N}, \;1-\frac12+\frac13-\frac14-\dotsb-\frac{1}{1318}+\frac{1}{1319}=\frac{p}{q}.\qquad \text{Prove that } 1979\mid p.$$ So my solution went like ...
7
votes
4answers
546 views

How to rewrite $7-\sqrt 5$ in root form without a minus sign?

How to rewrite $7-\sqrt 5$ in root form without a minus sign ? For clarity "root form " means an expression that only contains a finite amount of positive integers , additions , substractions , ...
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1answer
59 views

Finding a value that a set of given polynomials have the largest possible common divisor

Let $f_1, ..., f_n$ be homogenous polynomials in $Z[x_1, .., x_4]$. Find the value $\alpha=(\alpha_1, ... , \alpha_4)$ such that $f_1, ... , f_n$ evaluated at $\alpha$ have the largest common ...
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0answers
37 views

Extension of quadratic forms

A homogen multivariate polynomial with degree 2 is a quadratic form. It can be checked if the polynomial is positive for any non-zero vector by checking if the corresponding matrix A is positive ...
4
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0answers
31 views

Set of Metapolynomials is closed under multiplication

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form $$f(x_1,\cdots , x_k ...
0
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5answers
46 views

Fraction with negative exponent fraction.

Q: $$\left(\frac{27 a^6 b^{-3}}{c^{-2}}\right)^{-2/3}$$ A: $$\frac{b^2}{9 a^4 c^{4/3}}$$ How in the world are they getting that?
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0answers
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evaluation of polynomial regression

I have a data set $(x_i$ $y_i)$ if=1...20. I have to fit the data using polynomial feature. How can I evaluate what the complexity of model should be chosen? There is a hint in the task using RMSE ...
2
votes
1answer
23 views

Explicit delta for polynomial limit

I'm looking for an explicit formulation for $\delta$ in the $\epsilon-\delta$ formulation of the limit for a polynomial $p(x) = \sum_{n=0}^N a_nx^n$. For example, in the the specific linear case ...
0
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3answers
85 views

for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
0
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0answers
21 views

Derivation of composite Gaussian quadrature error formula

I am working on studying for the Numerical Analysis qualifying exams. One of the questions I am stuck on is the following: Derive the error term for the composite Gaussian quadrature rule with ...
0
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1answer
42 views

Polynomials - getting wrong answer using Euclidean algorithm

I am finding the GCD of $a = x^3 + 11/3x^2 + 17/4x + 3/2$ and $b = 3x^2 + 22/3x + 17/4$ using the Euclidean algorithm. So I divide $a/b$ and get $q$ and $r$ such that $a = qb + r$. Then, according to ...
9
votes
1answer
271 views

$\epsilon>0$ there is a polynomial $p$ such that $|f(x)-e^{-x}p|<\epsilon\forall x\in[0,\infty)$

Could any one tell me how to solve this one? Given $f\in C[0,\infty)$ such that $f(x)\to 0$ as $x\to\infty$ we need to show that for any $\epsilon>0$ there is a polynomial $p$ such that ...
11
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0answers
199 views

Polynomials with rational zeros

Find all polynomials $F(x)={a_n}{x^n}+\cdots+{a_1}x+a_0$ satisfying $a_n \neq0$; $(a_0, a_1, a_2, \ldots ,a_n)$ is a permutation of $(0, 1, 2 ... n)$; all zeros of $F(x)$ are rational.
4
votes
0answers
67 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
2
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1answer
272 views

Using the Multinomial Theorem to Calculate a Finite Sum raised to an exponent

I know it's a simple question, but I keep getting different general formulas for the coefficients when I am trying to use the multinomial theorem for the following: $$ ...
2
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0answers
39 views

How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
2
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1answer
103 views

Three phase voltage system of polynomial equations

I'm working with the development of a product in the company where I work. This product measures three phase voltages and currents. I cannot change the circuit because it has been sold for a long time ...
0
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3answers
42 views

In $\mathbb{Q}[x]$, what is the gcd of $x^6 − 1$ and $x^4 − 1$…Using Euclid's Method. [closed]

In $\mathbb{Q}[x]$, what is the gcd of $x^6 − 1$ and $x^4 − 1$...Using Euclid's Method. Could someone please do the first few steps of this so I know how to solve gcd of polynomials using Euclid's ...
2
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1answer
39 views

Efficient Extended GCD Algorithm for Polynomials

For computing the GCD of two multivariate polynomials we have the Euclidian algorithm. However, it's well known that the Euclidian algorithm is not very efficient (because of intermediate expression ...
0
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0answers
21 views

vector subspace of all real polynomials which are divisible with $x^2 + 1$

Show that the set of all real polynomials which are divisible with $x^2 + 1$ is a vector subspace of space of all real polynomials to 4th degree. Also find base and dimension of this subspace. I ...
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3answers
60 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
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0answers
107 views

Reduced Gröbner basis and extension of scalars

Consider a field extension $L\subseteq K$, and let $\mathfrak a\neq 0$ be an ideal of the polynomial ring $L[T_1,\ldots,T_n]$. Suppose that a monomial order is fixed, so there exists a unique reduced ...
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2answers
86 views

Find the roots of the equation $(1+xi)^n+(1-xi)^n=0$

Find the roots of the equation $f(x)=(1+xi)^n+(1-xi)^n=0$. I'm having problems finding the roots...this is what I've done: First I expressed $(1+xi)^n$ and $(1-xi)^n$ in trigonometric form and ...
0
votes
5answers
69 views

Graphing polynomials

Sketch a graph of the polynomial $P(x)=(x-2)^2(x+1)^3$. You must plot and label the x and y intercepts and these should be the only points you plot. How do I sketch the graph of a polynomial?
3
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0answers
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Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...