2
votes
1answer
79 views

How to find all complex polynomial $f$ such that $1+f(z^n+1)=(f(z))^n$

Question: Let $n\gt1$ be a natural number. Is there a non-constant complex polynomial $P$ such that $P(x^n+1)=P(x)^n-1$ for all $x$? I saw this problem about polynomial, here is the question: Find ...
0
votes
1answer
20 views

Factor the equation either by pairs method or any other [closed]

I tried to split by pairs but i got nowhere. $$x^2 - 4y^2 - 4x + 4$$
0
votes
2answers
26 views

Understanding Multiplicities

I am having troubles understanding what 'multiplicities' mean. In example what does $-1/3(multiplicity 2)$ translate into?? To clarify this is for finding zero's in a polynomial function Any help ...
0
votes
0answers
29 views

Polynomials that represent a function

Let $D(x,n_1,\dots,n_k) \in \mathbb{Z}[x,n_1,\dots,n_k]$ be a polynomial. Every such polynomial represents a semi-decidable property of natural numbers by $$P(x) :\equiv (\exists n_1,\dots,n_k)\ ...
1
vote
3answers
41 views

How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
5
votes
1answer
109 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
0
votes
1answer
45 views

Degree of Equations [closed]

A) Which variables in the formula $V = \pi r^2 h$ would you need to set as a constant in order to generate: a linear equation? a quadratic equation? B) How should $r$ and $h$ be ...
2
votes
3answers
40 views

A certain polynomial P(x) , $x\in R$ when divided by $x-a, x-b,x-c$ leaves the remainders a,b,c respectively…

A certain polynomial P(x) , $x\in R$ when divided by $x-a, x-b,x-c$ leaves the remainders a,b,c respectively. Find the remainder when P(x) is divided by $(x-a)(x-b)(x-c)$ is (a,b,c are distinct) My ...
1
vote
1answer
59 views

$f(n)$ and $f(2^n)$ are co prime for all natural numbers $n$. Find all such polynomials.

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^n)$ are co prime for all natural numbers $n$.
0
votes
2answers
43 views

Are there infinitely many polynomials crossing through a finite amount of points?

If I have $3$ $(x,y)$ points, say $(2,3)$, $(8,17)$ and $(20,25)$, how many polynomials are there that pass through those $3$ points? Infinitely many or a finite amount? What if I have $n$ arbitrary ...
0
votes
1answer
34 views

Linear independence of a set of 'simple' functions

I have some $\alpha = \{f_0(x),f_1(x),\dots,f_n(x)\}$ Where having the $k$ in $f_k(x)$ equal $x$, meaning $k=x$ makes $f_k(x)=1$ or zero if $k \ne x$. Now I want to show $\alpha$ is linearly ...
0
votes
1answer
23 views

write all functions as rational functions

my question is short and simple : could we write all functions that have the formula : $1 + G(s)H(s)$ as $$\frac{(s+b_0)(s+b_1)\dots(s+b_n)}{(s+p_0)\dots(s+p_m)}$$ if the answer is yes , could you ...
0
votes
2answers
30 views

Function tranlsation $g(x) = f(x) + 15$

I can't seem to work this answer out when practicing for exams. Here's the question: You are given that $f(x) = (2x - 3)(x + 2)(x + 4) \cdots$ From this I know $f(x)$'s roots: $\frac{3}{2}$, ...
0
votes
0answers
29 views

Approximate function algorithm using a polynomial and Boor splines

I have a defined function and a set of points with equal distance between them. The problem is that I have to approximate the graphic of that function using a polynomial function of 3rd degree and a ...
3
votes
1answer
51 views

Find all roots of the equation $1-\frac{x}{1}+\frac{x(x-1)}{2!}-\cdots+(-1)^n\frac{x(x-1)(x-2)…(x-n+1)}{n!}=0$

$ 1-\dfrac{x}{1}+\dfrac{x(x-1)}{2!}-\cdots+(-1)^n\dfrac{x(x-1)(x-2)...(x-n+1)}{n!}=0$ I could not attempt the question at all but just rewriting it as $P(x)=\dfrac{1}{n!}\prod\limits_{k=1}^n (k-x)$ ...
0
votes
2answers
15 views

Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ …

Problem : Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ and $f(0) =2+\sqrt{3}+\sqrt{5}$ Prove that the equation $f(2009)x^2 +2f(0)x ...
1
vote
1answer
44 views

Find $P(12)$ when $P(x)={1\over(x+1)}$, for $x=0,1,2,…11$

let $P(x)$ be a polynomial of degree $11$ such that $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$ then find value of $P(12)$
1
vote
0answers
22 views

Creating non-linear function to calculate points per distance in reversed order.

I have a small game in which I want to give points according to closeness to location, so the maximum points will be given for the minimal number = 1. It's similar to GeoGuesser game. Here is the data ...
1
vote
0answers
28 views

Formula for calculation score based on distance

I try to write function for calculating scoring from distance in my game. I found something similar : link But I need the distance(x) to be between 0-31855000 meters and score between 0 to 1000. ...
0
votes
1answer
31 views

Is there a way to compute if(i < j) k := (a + b)c with polynomials over $\Bbb{Z}_p$?

Let $p$ be a prime and let all variables be in $\Bbb{Z}_p$. Then you can write the result of if(i > 0) k = (a + b)c; (C code) as a polynomial $k := ...
1
vote
1answer
37 views

how to determine a function of a specific shape

I have become very passionate about mathematics lately, but one question is really irritating me, HOW do you determine a function for ANY specific shape ? Is there a certain procedure that enables ...
5
votes
4answers
106 views

Find$f$ s.t. $f(1)=2$, $f(2)= 4$, $f(3)= 6$ and $f(4)= \pi$.

Find a function where $f(1)=2$, $f(2)= 4$, $f(3)= 6$ and $f(4)= \pi$. I got $\dfrac16(x-3)(x-2)(x-1)\pi$ as a start to get rid of $\pi$.
1
vote
2answers
53 views

Describing asymptotic behaviour of a function

For question B! x^2+x+1/x^2 = 1+ [x+1/x^2] shouldnt the answer be asymptote at x=0 and y=1 ?? i dont understand the textbook solution
2
votes
2answers
67 views

Does there exist a polynomial function for every n points, whose extremas are these points?

Given $ n $ points in $ \mathbb{R}^2 $, does there exist a polynomial function of any degree, whose extremas include these $ n $ points? Given 3 points: $ P_1 = (0,4), P_2 = (2,2), P_3 = (4,7) $ And ...
1
vote
1answer
40 views

How to determine the remaining roots when two distinct real roots and y-intercept are given

A quartic polynomial has 2 distinct real roots at $x=1$ and $x=-3/5$. If the function has a y-intercept at -1 and has $f(2)=2$ and $f(3)=3$, determine the remaining roots and produce an accurate ...
0
votes
1answer
467 views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
0
votes
1answer
40 views

Inferring a characteristic of a ratio of functions from the ratio of their derivatives

This is a strange one, but I need help trying to understand whether there is any logic behind this or not. Given $\frac {f(\sqrt{2})}{g(\sqrt{2})}=2$, and $\frac {f'(x)}{g'(x)}>2$ for all ...
1
vote
4answers
58 views

Let $f(x)=x^2+17x+a$, $g(x)=x^2-17x-a$, $r$ a root of $f$ and $-r$ a root of $g$. Determine the roots of $f$.

Let $f(x)=x^2+17x+a$ and $g(x)=x^2-17x-a$. Suppose $r$ is a root of $f$ and $-r$ is a root of $g$. Determine all roots of $f$. From the descriptions, I can conclude that $f(x)-g(x)=2a$. But that ...
0
votes
2answers
237 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
0
votes
1answer
17 views

Inequality Conditions

Let $h_{k}(x)>0$ and $\sum_{k=1}^{l}h_{k}(x)=1$ (Here, $h_{k}(x)$ are some continuous functions). Is the statement below correct or not? $f_{k}(x)<0$ when $g_{k}(x)=0$, $\forall x \neq 0$, ...
0
votes
2answers
92 views

Cubic function that has no y intercept

Is there a cubic function that is only in quadrants 1 and 2 of the coordinate plane and so never crosses the y axis? If so can you give me a cubic function that does that?
2
votes
1answer
131 views

Degree of a function

I found on wikipedia (http://en.wikipedia.org/wiki/Degree_of_a_polynomial) that a degree of a general function can be computed as $$\deg f(x) = \lim_{x\to\infty}\frac{\log |f(x)|}{\log x}$$ or $$\deg ...
2
votes
2answers
204 views

Prove, that two equations are equivalent

EDIT: Missed something very important! Sorry! We have $x^4+1=2(2x-1)^{1/4}$ not $x^4+1=2\sqrt{2x-1}$. One friend of mine told me that the equation $x^4+1=2(2x-1)^{1/4}$, where $x\geq \frac{1}{2}$ is ...
0
votes
0answers
26 views

Solution of a partial fraction equation

Consider the equation in the variable $\lambda$ \begin{align} \sum_{i=1}^{N}\frac{a_i}{(\lambda+b_i)^2}=1 \end{align} where $a_i$ and $b_i$ are all positive. How do you find any solution to this ...
2
votes
1answer
80 views

Find $g(x)$ if $f(g(x))=f(x)g(x)$ and $g(2)$=37, $f(x)$ and $g(x)$ are polynomials

Suppose $f(x)$ and $g(x)$ are non-zero polynomials with real coefficients, such that $f(g(x))=f(x)\times g(x)$. If $g(2)=37$, find $g(x)$. I tried plugging $f(x)$ and $g(x)$ as $n$ and $m$ ...
0
votes
2answers
35 views

Get polynomial function from 3 points

I need to understand how to define a polynomial function from 3 given points. Everything I found on the web so far is either too complicated or the reversed way around. (how to get points with a given ...
0
votes
0answers
25 views

Convert function form to variate polynomials form

Let $f:{\Bbb Z}\longrightarrow{\Bbb Z}$ be a function, example $f(0)=3,\ f(1)=2,\ f(2)=0,\ f(3)=1$. Now that was "in decimal". We can think of it "in binary" format: $f(00)=11,\ f(01)=10,\ f(10)=00,\ ...
0
votes
1answer
58 views

Spivak Chapter 3 Question 3. [duplicate]

Working through Chapter 3 of Spivak's Calculus. The question is: If $x_1, x_2, \ldots, x_n$ are distinct numbers, find a polynomial function $f_i$ of degree $n-1$ which is 1 at $x,$ and 0 at $x_j$ ...
1
vote
2answers
94 views

Apostol Calculus Vol.1 Exercise 9 , Chapter 1.5 (Prove property of polynomial function)

Ok, so I have a huge problem with this exercise. It is a property of polynomial functions that needs proving. Thing is, I can not even get a clue and I put in some numbers and it does not seem to ...
8
votes
4answers
157 views

Why isn't $y=(x^6)^{1/3}$ a polynomial function?

I've been told that $y=(x^6)^{1/3}$ isn't a polynomial function because of the radical but I believe that the equation could be simplified to $y=x^2$ which fits the definition of a polynomial ...
4
votes
2answers
82 views

How is called the class of functions whose inverse function is a polynomial?

How is called the class of functions whose inverse function is a polynomial? Is there any study of such functions?
0
votes
4answers
125 views

Check that two function $f(x,y)$ and $g(x,y)$ are identical

Given that $f(x)$ and $g(x)$ are two polynomials of degree $n$, we know that if we can find $n+1$ distinct numbers $x_i$, $i=1,\cdots,n+1$ such that $f(x_i)=g(x_i)$ then $f(x)$ and $g(x)$ are ...
3
votes
1answer
74 views

Show that T is a linear transformation and find a, b, c

I'm having trouble understanding this question and the proper way to solve it. I don't understand the solution given and why this was the right way to answer it. Problem: For the vector space ...
1
vote
1answer
95 views

A polynomial is called a Fermat's polynomial…

A polynomial is called a Fermat polynomial if it can be written as the sum of the squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0) = ...
0
votes
1answer
20 views

given the polynomial function find the function values

I given a problem and I can not get to the answer that I was given by my professor. P(x) = -x^2 - x + 12 where P(-2). I keep coming up with 10. He gave me the answer of -2. Am I missing something?
3
votes
1answer
295 views

Finding roots of a function in an interval

Does the equation $x^3-12x+2=0$ have three solutions in the interval $[-4,4]$? We know that this is a continuous function because it's a polynomial, and so we can use the Intermediate Value ...
0
votes
0answers
47 views

Conversion of roots of a polynomial

I'm wondering, given a polynomial $P(x)$ with roots $r_i (1\le i\le n)$, how to determine the polynomial $Q(x)$ such that its roots are $r'_i=f(r_i)$. For example, if $P(x)=x^2-x-6=(x-3)(x+2)$ and ...
0
votes
1answer
53 views

A physics related question about an infinitely long pipe.

This is a really nice question I found some days ago, so I translated it into English to share. Suppose we have a water pipe which is infinitely long, with water flowing in it. We know that if a ...
4
votes
2answers
175 views

What is a polynomial and how is it different from a function?

I have a problem that asks me to find a polynomial $P(x)$ so that $P(3)$ is 9. Now I can say with certainty that $P(x)$ can be $x^2$. This is a second degree polynomial. But what about functions ...
3
votes
2answers
55 views

Find the equation with roots, $A$, $B$, $C$ is $ABC=6$, $A+B+C=5$ and $A^2 +B^2+C^2=21$

Find the equation with roots, $A$, $B$, $C$ is $ABC=6$, $A+B+C=5$ and $A^2 +B^2+C^2=21$ Can someone please hint me, or show me what do i do with this question please. Im quite clueless and need to be ...