In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.
Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.
Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely,
$$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$,
It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...
The Homework Statement reads the question.
We have
$$
\langle f,g \rangle = \sum_{k=0}^{n} f\left(\frac{k}{n}\right) ~g\left( \frac{k}{n} \right)
$$
If ##f(t) = t##, we have degree of ##f## is ##1##, so, should I take ##n = 1## in the above inner product formula and proceed as follows
$$...
This is exercise 12.1.2 a from Arfken's Mathematical Methods for Physicists 7th edition :
Starting from the Laguerre ODE,
$$xy''+(1-x)y'+\lambda y =0$$
obtain the Rodrigues formula for its polynomial solutions $$L_n (x)$$
According to Arfken (equation 12.9 ,chapter 12) the Rodrigues...
I could simplify the expressions in the numerator and denominator to (1+x^n)/(1+x) as they are in geometric series and I used the geometric sum formula to reduce it. Now for what value of n will it be a polynomial?
I do get the idea for some value of n the simplified numerator will contain the...
Hello everyone. I need to construct in python a function which returns the evaluation of a Chebishev polynomial of order k evaluated in x. I have tested the function chebval form these documents, but it doesn't provide what I look for, since I have tested the third one, 4t^3-3t and
import numpy...
i have some doubts from chapter 1 of the book Mathematical methods for physics and engineering.
i have attached 2 screenshots to exactly point my doubts.
in the first screenshot...could you tell me why exactly the 3 values of f(x) are equal.
the first is the very definition of polynomials...but...
I was reading this book - " mathematical methods for physics and engineering"
in it in chapter 1 its says
"F(x) = A(x - α1)(x - α2) · · · (x - αr),"
this makes sense to me but then it also said
We next note that the condition f(αk) = 0 for k = 1, 2, . . . , r, could also be met
if (1.8) were...
I'm reading a book where the author gives the long division solution of ##\frac 1 {1+y^2}## as ##1-y^2+y^4-y^6...##. I'm having trouble duplicating this result and even online calculators such as Symbolab are not helpful. Can anyone explain how to get it?
Homework Statement
##3f(x)+2f(\frac{1}{x}) = x##, solve ##f(x)##
Homework Equations
Not sure.Maybe the ones of inverse functions.
The Attempt at a Solution
The only thing that I came up so far is that the function’s highest order term is ##x## because if there are higher orders,it will show...
Recently I came up with a proof of “ for a nth degree polynomial, there will be n roots”
Since the derivative of a point will only be 0 on the vertex of that function,and a nth degree function, suppose ##f(x)##has n-1 vertexes, ##f’(x)## must have n-1 roots.
Is the proof valid?
$$\int x^2+3 = \frac{x^3}{3}+3x+C$$
I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions:-p
Homework Statement
Define {x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!} for positive integer n. For what values of positive integers n and m is g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}} a factor of f(x)={{{x+1} \choose n} \choose {m}}?
Homework Equations
The idea...
Homework Statement
Find all ##a,b,c\in\mathbb{R}## for which the zeros of the polynomial ##az^3+z^2+bz+c=0## are in this relation $$z_1^3+z_2^3+z_3^3=3z_1z_2z_3$$
Homework Equations
we know that if we have a polynomial of degree 3 the zeroes have relation in this case
##z_1+z_2+z_3=-1/a##...
First time in this forum, so greetings to everyone!
I am currently working with some physical models in the field of natural ventilation and I came across the following 5th order polynomial equation (quintic function):
$X^{5}+ C X - C =0$
This is the steady state solution of a physical system...
Homework Statement
>Find the sum of the roots, real and non-real, of the equation x^{2001}+\left(\frac 12-x\right)^{2001}=0, given that there are no multiple roots.
While trying to solve the above problem (AIME 2001, Problem 3), I came across three solutions on...
Homework Statement
Hello,
I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space.
Homework Equations
- Knowledge of power series, polynomials, Legenedre...
Consider the summation ∑,i=0,n (t^(n-i))*e^(-st) evaluated from zero to infinity.
You could break down the sum into: (t^(n))*e + (t^(n-1))*e + (t^(n-1))*e + ... + (t^(n-n))*e ; where e = e^(-st)
To evaluate this, notice that all terms will go to zero when evaluated at infinity
However, when...
All variables and given/known data and Relevant equations:
So I got the functions for a bottle design (one side with the bottle lying horizontally):
1. y=-1/343x^3+3/98x^2 + 2.5 ; 0<x<7
2. y=3; 7<x<15
3. y=-1/98x^2+15/49x+69/98; 15<x<22
Combined they give the volume of 570.2mL using the volume...
Homework Statement
[/B]
I am to design a 600mL water bottle by drawing one side (bottle lying horizontally). Three types of functions must be included (different orders). The cross-sectional view would be centred about the x-axis, and the y-axis would represent the radius of that particular...
Hello
I have a third order polynomial, for example y(x) = -60000x^3 - 260x^2 + 780x + 0.6
I need to know what is x at y = 28 and/or y= 32.
I can goto MATLAB and find the roots ( x = - .1158, -.0007, and .1122 )
or I can go to
http://www.wolframalpha.com
and it also finds the roots and...
Hello.
Assume that I have two polynomials of degree 2, i.e., Quadratic Equations.
1.
Assume that the Quadratic Equation is:
x2 + 7x + 12 = 0
The roots of the Quadratic Equation is -3 and -4.
2.
Assume that there is another Quadratic Equation:
x2 + 8x + 12 = 0
The roots of the Quadratic...
Hello everyone.
Iam working on a course in digital control systems and by reading my textbook I stumbled over this expression.
C(z) = 0.3678z + 0.2644 : z^2 − 1.3678z + 0.3678
= 0.3678z^−1 + 0.7675z^−2 + 0.9145z^−3 + ...
Now Iam wondering how the result of the polynomial division is...
Homework Statement
Algebra - I.M. Gelfand, Problem 164. Prove that a polynomial of degree not exceeding 2 is defined uniquely by three of its values.
This means that if P(x) and Q(x) are polynomials of degree not exceeding 2 and P(x1) = Q(x1), P(x2) = Q(x2), P(x3) = Q(x3) for three different...
Homework Statement
Determine if the following is a subspace of ##P_3##.
All polynomials ##a_0+a_1x+a_2x^2+a_3x^3## for which ##a_0+a_1+a_2+a_3=0##
Homework Equations
use closure of addition and scalar multiplication
The Attempt at a Solution
Let ##P=a_0+a_1x+a_2x^2+a_3x^3## and...
Homework Statement
Given the polynomial function ##x^4+x^3+2x^2+4=0## solve it if you know that it has at least one complex zero whose real part equals the complex part.
Homework Equations
3. The Attempt at a Solution [/B]
My guess is that if this function has one complex zero it must have a...
I'm currently studying the sensitivity of polynomial roots as a function of coefficient errors. Essentially, small coefficient errors of high order polynomials can lead to dramatic errors in root locations.
Referring to the Wilkinson polynomial wikipedia page right...
Homework Statement
Help i have a homework quiz done and i simply can't find out how to do the 3rd problem as we haven't even learned how to do it or maybe my notes aren't good or something , however I am close to an A in the class and this would help bring it closer. It asks me: "Find all the...
Homework Statement
Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms.
Degree 4: No real zeros, complex zeros of 1+i and 2-3i
Homework Equations
(-b±√b^2-4ac)/2a
The Attempt at a Solution
I want...
NOTE: presume real coefficients
If a pair of polynomials have the Greatest Common Factor (GCF) as 1, it would seem that any root of one of the pair cannot possibly be a root of the other, and vice-versa, since as per the Fundamental Theorem of Algebra, any polynomial can be decomposed into a...
Hello, I have a question regarding "polynomials" that have terms with interger and fractional powers.
Homework Statement
I want to solve:
$$ x+a(x^2-b)^{1/2}+c=0$$
Homework Equations
The Attempt at a Solution
My approach is to make a change of variable x=f(y) to get a true polynomial (integer...
Homework Statement
Prove the following statement:
Let f be a polynomial, which can be written in the form
fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0
and also in the form
fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0
Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n
Homework Equations
3. The...
Homework Statement
[/B]
Th value of 'a' for which the equation x3+ax+1=0 and x4+ax+1=0 have a common root is?
Homework Equations
The Attempt at a Solution
i initially thought of subtracting both the equations and then finding x and substituting back in the equation but it did not work.
Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This...
Dear PF Forum,
As we know in polynomial 2 degrees AX2 + BX + C = 0, there's a formula for solving it.
What about 3 degrees for example: AX3 + BX2 + CX + D = 0, there's is really no formula for solving it?
The only way to solve it is by hand?
I have several methods in my head, at least...
Homework Statement
How many pairs of solutions make x^4 + px^2 + q = 0 divisable by x^2 + px + q = 0
Homework Equations
x1 + x2 = -p
x1*x2= q[/B]
The Attempt at a Solution
I tried making z = x^2 and replacing but got nowhere. I figure 0,1,-1 are 3 numbers that fit but I am not sure what's...
Just to double check, but if one wanted to, like in partial fraction decomposition, associate literal coefficients of polynomials with corresponding unknowns on the other side of the equation, the justification for this action is the definition of equality of polynomials?
EDIT: I know this...
I have been given a task to create an interpolating/extrapolating programme. I have completed the programme for linear interpolation (2 points) but now must make it usable for 3 or more points, ie a polynomial of n points. I think I have the equation in general for a polynomial as it is an...
I was reviewing the Cardano's method formula for a real cubic polynomial having 3 real roots. It seems that to do so, the arccos (or another arc*) of a term involving the p & q parameters of the reduced cubes must be done, and then followed by cos & sin of 1/3 of the result from that arccos -...
Umm from memory I used to use...that triangle:
1
1 1
1 2 1
1 3 3 1
Fibonachii was it? Pathetic I can't even remember the name.
To factorise...or was it expand...polynomials...anyway, I don't think that's elevant here.
My question is; I had an...
I am trying to use a numerical polynomial root finding method, but I am unsure of the order of an expression. For example, if I have something that looks like
x2+5x √(x2+3)+x+1=0
what is the coefficient of the second order (and potentially even the first order) term? Is the entire 5x√... term...
Homework Statement
Show that if
P(z)=a_0+a_1z+\cdots+a_nz^n
is a polynomial of degree n where n\geq1 then there exists some positive number R such that
|P(z)|>\frac{|a_n||z|^n}{2}
for each value of z such that |z|>R
Homework Equations
Not sure.
The Attempt at a Solution
I've tried dividing...
Can someone just confirm my answers to this easy polynomial question,
State the degree and dominant term to f(x)=2x(x-3)^3(x-1)(4x-2)
I am working on this online and there is nothing on working on equations like this in the lesson. I believe the degree to be either 2 or 6, as the functions end...