What is Polynomial: Definition and 1000 Discussions

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.

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  1. N

    Taylor Polynomial of f(x) = x^3sin(x)

    Homework Statement Find the first 3 non-zero terms of the Taylor polynomial generated by f (x) = x^{3} sin(x) at a = 0. Homework Equations f^{n}(x) * (x-a)^{n} / (n!) The Attempt at a Solution I got the question wrong: my answer was 1/3! + 1/5! + 1/7! Here is the answer below. I...
  2. J

    MHB Polynomial of Degree 98: Value of $p(100)$

    If $p(x)$ be a polynomial of degree $98$ such that $\displaystyle p(x) = \frac{1}{x}$ for $x=1,2,3,...,98$ Then value of $p(100)=$
  3. C

    Find sum of roots of polynomial

    How would I go about approaching this problem? Given the polynomial: x^100 - 3x + 2 = 0 Find the sum 1 + x + x^2 + ... + x^99 for each possible value of x.
  4. A

    Integral of an exponential that has a polynomial

    How would one evaluate $$\Phi = \int_{-\infty}^{+\infty} e^{-(ax+bx^2)} dx$$. I was trying to change it into a product of an error function and a gamma function, but I needed an extra dx. Any other ideas?
  5. J

    MHB Roots of a polynomial with non-real coefficients.

    Let a,b,c,d be real numbers. Sauppose that all the roots of the equation $z^4 + az^3 + bz^2 + cz + d = 0$ are complex numbers lying on the circle $\mid z\mid = 1$ in the complex plane. The sum of the reciprocals of the roots is necessarily: options a) a b) b c) -c d) d ---------- Post...
  6. X

    Turing machine - polynomial time expression

    In the Turing Machine, the machine accepts a word if the computation terminates in the accepting state. The language accepted by the machine, L(M), has associated an alphabet Δ and is defined by L(M) = {w \in \Delta} This means that the machine understands the word w if w belongs to the...
  7. caffeinemachine

    MHB Polynomial of degree 3. splitting field.

    If $F$ is the field of rational numbers, find the necessary and sufficient conditions on $a$ and $b$ so that the splitting field of $p(x)=x^3+ax+b=0$ has degree exactly $3$ over $F$. ATTEMPT: If $p(x)$ is not irreducible in $F[x]$ then the splitting field of $p(x)$ over $F$ can have degree...
  8. H

    Factorization of a complex polynomial

    Homework Statement p(x)=((x−1)^2 −2)^2 +3. From here find the full factorization of p(x) into the product of first order terms and identify all the complex roots. Homework Equations I am having trouble doing this by hand. I know there are four complex roots but can't seem to figure out...
  9. S

    MHB Polynomial Function: Find All Zeros (Real & Complex)

    Polynomial function f(x)= x^3-12x^2+46x-52 A. List possible rational ZerosB. find all the zeros (real and complex) of the function (test x=2 as a rational zero using the synthetic division?
  10. G

    What are Ideals and Polynomial Rings? A Simple Explanation and Concrete Example

    Can anyone explain ideals and polynomial rings i.e. definitions, examples, the most important theorems, etc.?
  11. J

    MHB Minimum degree of a polynomial passing through points

    If p(x) is a polynomial such that p(0)=5 ,p(1)=4 ,p(2)=9,p(3)=20 , the minimum degree it can have
  12. J

    How do I factor 4 term polynomial?

    Hi guys, I'm not entirely good with factoring so I was wondering if any of you can show me how I would go about factoring this polynomial: (With necessary steps if you can Please and thanks!) 36mx + 10y - 24x - 15my
  13. S

    Solving polynomial equations,hlep

    we assume a,b,c,d are unknown variables,whose solution are either 0 or 1.So a power of a variable equals to itself(e.g,a^(n)=a).Would u please help me find a proper way to sovle the following simultaneous equations whose solutions are either 0 or 1...
  14. H

    Cubic polynomial function with 3 real roots; one at infinity?

    Is it possible to have a cubic polynomial (ax^3+bx^2+cx+d) which has three REAL roots, with one of them being +/- infinity? If there is, can you give an example? Thanks!
  15. N

    Partial Fraction Question: HELP with Polynomial Long Division

    Homework Statement x^2-x-13/(x^2+7)(x-2) hello i am having trouble solving this problem.. could anyone please show me how to do this step by step? i know polynomial long division is required before it can be converted to partial fractions. I also know the answer is 2x+3/x^2+7 - 1/x-2...
  16. L

    Binomial expansion comparison with legendre polynomial expansion

    Hi, I've been working on this question which asks to show that {{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}{d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}} So first taking the n derivatives of the binomial expansions of (x2-1)n...
  17. B

    MHB Linear independence of polynomial set.

    Hi guys, I've been working on a question which is as follows: For which real values of c will the set $\{1+cx, 1+cx^2, x-x^2\}$ be a basis for $P_2$? I'm coming up with the answer as no values of c, but am I really wrong? I've only checked linear independence, because it would imply that it...
  18. F

    Proof that e^z is not a finite polynomial

    Homework Statement Prove that the analytic function e^z is not a polynomial (of finite degree) in the complex variable z. The Attempt at a Solution The gist of what I have so far is suppose it was a finite polynomial then by the fundamental theorem of algebra it must have at least...
  19. P

    Eigenvalue of Polynomial Transformation

    Homework Statement Let T:P2→P2 be defined by T(a0+a1x+a2x2)=(2a0-a1+3a2)+(4a0-5a1)x + (a1+2a2)x2 1) Find the eigenvalues of T 2) Find the bases for the eigenspaces of T. I believe the 'a' values are constants. Homework Equations None. The Attempt at a Solution The problem I am...
  20. B

    Roots of a polynomial and differenciaton

    Homework Statement I read that if f'(x) is zero once in [a b] then f(x) has maximum two real roots. Why maximum? Shouldn't it be exactly 2? Or it has something to do with the case of repeated roots? Homework Equations The Attempt at a Solution was thinking as in figure
  21. M

    Proof that if a polynomial has a complex zero it's conjugate is also a zero

    Homework Statement If P(x) is a polynomial with real coefficients, then if z is a complex zero of P(x), then the complex conjugate \bar{z} is also a zero of P(x). Homework Equations Book provides a hint: assume that z is a zero for P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} and...
  22. S

    Complex polynomial properties when bounded (Liouville theorem)

    Homework Statement Suppose f is differentiable in \mathbb{C} and |f(z)| \leq C|z|^m for some m \geq 1, C > 0 and all z \in \mathbb{C} , show that; f(z) = a_1z + a_2 z^2 + a_3 z^3 + ... a_m z^m Homework EquationsThe Attempt at a Solution I can't seem to show this. It does the proof...
  23. S

    How Do You Solve a Complex Polynomial and Trigonometry Problem?

    Homework Statement If x3 + 5x2 + 4x = 3 = 0 and cos (5 - 3x) = √p, find the value of cot (x5 + 2x4 - 6x3 + 16x2 + 8x + 20) Homework Equations trigonometry polynomial The Attempt at a Solution stuck from the beginning...:-p
  24. M

    Proof that a polynomial is a factor

    Homework Statement Show that x+a is a factor of x^{n}+a^{n}for all odd n. The Attempt at a Solution (1) Assume that x+a is a factor of x^{n}+a^{n}for all odd n. This implies that when x^{n}+a^{n} is divided by x+a the remainder is zero. I don't know - is this a sensible 1st step...
  25. P

    Show that Characteristic polynomial = minimal polynomial

    Homework Statement Let A = \begin{pmatrix}1 & 1 & 0 & 0\\-1 & -1 & 0 & 0\\-2 & -2 & 2 & 1\\ 1 & 1 & -1 & 0 \end{pmatrix} The characteristic polynomial is f(x)=x^2(x-1)^2. Show that f(x) is also the minimal polynomial of A. Method 1: Find v having degree 4. Method 2: Find a vector v of...
  26. L

    Prove Polynomials Can be Written Using the Dimension Theorem

    Use the dimension theorem to show that every polynomial p(x) in Pn can be written in the form p(x)=q(x+1)-q(x) for some polynomial q(x) in Pn+1. I need to see all the steps so that I understand how to do it. PLease and Thank you
  27. C

    Weirdness of polynomial long division algorithm

    "Weirdness" of polynomial long division algorithm Hello. So, i just started to learn about the polynomial long division. As an introductory example, the book presents the long division of natural numbers, claiming that its basically the same thing. The example: 8096:23 Solution...
  28. L

    A general condition on polynomial roots

    Consider a polynomial of the following type: A_n w^n + A_{n-1} w^{n-1}k + A_{n-2} w^{n-2} k^2 + ... + A_1 k^n =0 What are the general conditions on {A_i} in order for the roots w(k) to be EITHER real OR functions with even imaginary parts, Im[w[k]]=Im[w[-k]]? I would be interested in...
  29. P

    MHB Proving Irreducibility of x^4 −7 Using Polynomial Theorems

    Explain why the polynomial x^4 −7 is irreducible over Q, quoting any theorems you use.
  30. U

    Verifying Linear Polynomial Mapping

    Homework Statement Prove whether the below equations are linear or not. (iii) U = P^2 -> V = P^6; (Tp)(t) = (t^2)p(t^2) + p(1). (iv) U=P^2 -> V =P^6;(Tp)(t)=(t^2)p(t^2)+1. Homework Equations None. The Attempt at a Solution I really don't know. Thanks Tom
  31. K

    Generate hermite polynomial coefficients

    Homework Statement I need to generate coefficients of hermite polynomials up to order k. Recursion will be used. Homework Equations a[n+1][k] = 2a[n][k-1] - 2na[n-1][k] The Attempt at a Solution Its not pretty, but here is my code. #include <iostream> #include <iomanip>...
  32. T

    Legendre Polynomial (anti)symmetry proof

    Homework Statement Let P_{n}(x) denote the Legendre polynomial of degree n, n = 0, 1, 2, ... . Using the formula for the generating function for the sequence of Legendre polynomials, show that: P_{n}(-x) = (-1)^{n}P_{n}(x) for any x \in [-1, 1], n = 0, 1, 2, ... . Homework Equations...
  33. E

    Polynomial functions and calculating dimensions

    Maria designed a rectangular storage unit with dimensions 1m by 2m by 4m. By what should he increase each dimension to produce an actual storage that is 9 times the volume of his scale model? v= (1) (2) (4) v= 8 v has to be 9 times larger v= (x+1) (x+2) (x+4) How do i find the value of x?
  34. E

    Determine the factor of a polynomial equation including piecewise functions

    The height,h, in meters, of a weather balloon above the ground after t seconds can be modeled by the function h(t)=-2t^3 + 3t^2 +149t + 410 for 0< t < 10. When is the balloon exactly 980m above the ground? 980 = -2t3 + 3t2 +149t + 410 0 = -2t3 + 3t2 +149t - 570
  35. F

    Nilpotent operator or not given characteristic polynomial?

    Hey, I'm working on a proof for a research-related assignment. I posted it under homework, but it's a little abstract and I was hoping someone on this forum might have some advice: Homework Statement Suppose T:V \rightarrow V has characteristic polynomial p_{T}(t) = (-1)^{n}t^n. (a) Are...
  36. F

    Nilpotent operator or not given characteristic polynomial?

    Homework Statement Suppose T:V \rightarrow V has characteristic polynomial p_{T}(t) = (-1)^{n}t^n. (a) Are all such operators nilpotent? Prove or give a counterexample. (b) Does the nature of the ground field \textbf{F} matter in answering this question? Homework Equations Nilpotent...
  37. J

    Qube root of 2, zero of second order polynomial

    How do you prove that there does not exist numbers a,b\in\mathbb{Q} such that 0 = a + b\sqrt[3]{2} + \sqrt[3]{2}^2
  38. N

    Simple Polynomial Factorization

    There is a theorem in algebra, whose name I don't recall, that states that given a polynomial and its roots I can easily factor it so for instance : p(x)=x^2-36 , assuming that p(x) is a real function, p(0)=0 \Leftrightarrow x=6,-6 then p(x) can be written as : P(x)=(x-6)(x+6) I...
  39. anemone

    MHB Prime number and the coefficients of polynomial

    Hi, I've got an equation stating p=a(r-1). If p represents prime number and r is a positive integer, and a is a constant, what can we conclude for the constant a? Like a $\in${-1, 1, -p, p}? I suspect this has something to do with modular arithmetic...:confused: Thanks.
  40. O

    Taylor polynomial of degree 1 - solve for theta

    Homework Statement I was given the following problem, but I am having a hard time interpreting what some parts mean. We're given the equation sinθ+b(1+cos^2(θ)+cos(θ))=0 Assume that this equation defines θ as a function, θ(b), of b near (0,0). Computer the Taylor polynomial of...
  41. J

    How to factor a polynomial modulo p?

    I can understand most of Galois Theory and Number Theory dealing with factorization and extension fields, but I always run into problems that involve factorization mod p, which I can't seem to figure out how to do. I can't find any notes anywhere either, so I was wondering if someone could give...
  42. J

    Splitting field of irreducible polynomial

    I need to find the splitting field in \mathbb {C} of x^3+3x^2+3x-4 (over \mathbb{Q} ). Now, I plugged this into a CAS and found that it is (probably) not solvable by radicals. I know that if I can find a map from this irreducible polynomial to another irreducible polynomial of the same...
  43. N

    How Do You Solve the Cubic Equation x^3 - 10x + 18 = 0?

    Homework Statement Hello there! I'm trying to find the roots of the following cubic polynomial x^3 - 10x + 18 = 0 The Attempt at a Solution I did the following: I rewrite 18 as 18 = - (x^3 - 10x) then I did back substitution and factored out x^3 - 10x - x^3 + 10x = 0 or x(x^2-10) -...
  44. K

    Fourier approximation with polynomial

    Homework Statement Approximate the function f(x)=sin(\pi x) on the interval [0,1] with the polynomial ax^{2}+bx+c with finding a, b and c. Homework Equations f(x)=a_{0}+\sum^{\infty}_{n=1}(a_{n}cos(nx)+b_{n}sin(nx)) a_0=\frac{1}{2\pi}\int^{\pi}_{-\pi}f(x)dx...
  45. anemone

    MHB Probability concerning polynomial.

    Let A, B, C be random number between (0,1). What is the probability that the polynomial Ax^2+Bx+C=0 has no real roots? I know that this question is a kind of c.r.v problem (uniform distribution). Also, it has something to do with exponential random variables. My problem is, exponential random...
  46. F

    Polynomial Rings (Units and Zero divisors)

    Hi all, I would just like to get some clarity on units and zero-divisors in rings of polynomials. If I take a ring of Integers, Z4, (integers modulo 4) then I believe the units are 1 & 3. And the zero-divisor is 2. Units 1*1 = 1 3*3 = 9 = 1 Zero divisor 2*2 = 4 = 0 Now, If I...
  47. E

    Magnitude of Complex Exponential Polynomial Inequality

    Homework Statement Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem. Find b, such that maximum of the magnitude of the frequency response function...
  48. M

    Was Polynomial Zeros' Practical Application Studied?

    I apologize for the rather vague title. It's space-limited and I'm not sure how to concisely state what I want to know. Basically, I understand that the solutions to quadratic equations (and if I remember correctly cubic equations) often had surveying problems land surveying. However, quartic...
  49. Z

    Irreducible polynomial over finite field

    Homework Statement Factor x^16-x over the fields F4 and F8 Homework Equations factored over Z (or Q), x^16-x = (x*(x - 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) The Attempt at a Solution I know the that quadratic and higher terms I have left...
  50. D

    Factoring polynomial through grouping

    Homework Statement 2n - 6m + 5n^2 - 15mn Homework Equations No particular equation since this is factoring The Attempt at a Solution Keep in mind that I struggle when it comes to grouping as I'm not sure where I'm supposed to start but... 2n - 6m + 5n^2 - 15mn Group first 2...
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