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. U

    Interpreting results of a polynomial fit

    I am currently working on a gamma ray spectroscopy lab in which i have just fit a polynomial to my calibration points. The calibration points are in a relatively straight line, from x=40 to x=450, and y=34 to y=1300 for the first and last end points respectively. Where X is channel number, and Y...
  2. S

    MATLAB How to Generate Symbolic Polynomial in MATLAB with Given Parameters?

    I am trying to get MATLAB to be able to produce a symbolic polynomial and am having some issues. The polynomial I would like it to produce is given a d and k, X(k+1)-X(k-d)-X(k-d-1)-...-1 What I have so far is k = 5; d = 3; syms X; i = 0; while (k-(d+i) >= 0) terms(i+1) =...
  3. K

    What is the simplest way of selecting the last N terms of a polynomial?

    If you have a polynomial like (1+x)^6 = x^6+6 x^5+15 x^4+20 x^3+15 x^2+6 x+1, What function would you use to pick only the last N terms? For instance, for N=3 pick x^6+6 x^5+15 x^4 I've being using sum of a binomial times something, but was wondering if there is anything more simple. Thanks!
  4. B

    Ode 2nd degree with a polynomial as a coefficient

    Hi friends, I have been trying to solve the ode of second degree below with respect to z: d2y/dz2=(i/a*z+b)*y i is the complex i, a and b are constants i ended up with the summation of bessel functions of first end second kind. Then I checked with MATLAB ode solver it gives no...
  5. D

    Is xp(x) in the Range of T for Given Polynomials?

    Homework Statement Let T: P2-P3 be the linear transformation defined by T(p(x)) = xp(x). Which of the following are in R(T)? (a) x2 (b) 0 (c) 1 + x Homework Equations R(T) is the the set of all vectors in P3 which are images under T of vectors in P2. The Attempt at a Solution...
  6. N

    Finding all zeros of a Polynomial

    Homework Statement Find all zeros of k(x) = 4x^4-7x^2+3x Homework Equations n/a The Attempt at a Solution I don't know if I should find the factors of 0 and 4.. Can you help me what to start out with.. I can do other ones like p(x) = 3x^3-37x^2+84x-24. But this problem k(x) =...
  7. D

    Need help plotting a 4th order polynomial

    I need help from someone with a graphing calculator. My little brother lost my graphing calculator and synthetic division I can't make work becasue the terms aren't exactly friendly. The equation is, 24.059025x^4 + 94.98287069x^3 + 44.56696001x^2 - 827.1103204x = -1067.832496 The more...
  8. S

    Chebshev polynomial approximation

    Homework Statement Hi every body I am triyng to find a polynolial approximation to the function: f(x)= (x+2)ln(x+2) using the chebyshev polynomials, the idea is to use MATLAB to find the coeefficients of the approximation poly. using the comand double(int(...)) but this command...
  9. X

    Lagrange Polynomial Interpolation

    Homework Statement Find the polynomial p(x) of degree 20 satisfying: p(-10) =p(-9) = p(-8) = ...=p(-1) = 0 p(0) = 1 p(1) = p(2) = p(3) = ...p(10) = 0 Homework Equations L(x) := \sum_{j=0}^{k} y_j \ell_j(x) The Attempt at a Solution i tried using the formula above: a =...
  10. DryRun

    Find all complex roots of polynomial

    Homework Statement Given 2 + 3i is one root of the equation: x^4 - 6x^3 + 26x^2 -46x +65 = 0 Find the remaining roots. The attempt at a solution I am thinking about this as a possible solution although it is too long to be plausible, unless I'm wrong. let x = a + bi and then replace in the...
  11. J

    Characteristic polynomial coefficients

    For any characteristic polynomial determined from A - eI (where A is a nxn matrix, e is an eigenvalue and I is the identity matrix), is it a rule that the coefficient associated with the char. polynomail term of highest degree must be positive ? My tutor made a theory that if the...
  12. A

    Polynomial Functions (word problem)

    Homework Statement A quartic function has x-intercepts of -5, -1, 2, and 4. The point (3.-8) lies on the curve. Determine an equation of the function. Homework Equations The Attempt at a Solution y = k (x+5)(x+1)(x-2)(x-4) Expanded form: f(x)=kx^4 + 9x^2 - 38x + 40 <---My...
  13. W

    Solving Polynomial Inequalities Using Synthetic Division and the PQ Rule

    Homework Statement 4x5-16x4+9x3+23x2-15x-9 > 0 Homework Equations Synthetic division PQ Rule? The Attempt at a Solution Don't know how or where to begin
  14. D

    Can Real Coefficients Affect the Roots of a Polynomial?

    Homework Statement Let z^n + \sum_{k=0}^{n-1}a_kz^k be a polynomial with real coefficients a_k\in[0,1]. If z_0 is a root, prove that Re(z_0) < 0 or |z_0| < \frac{1+\sqrt{5}}{2}. Homework Equations The Attempt at a Solution I have attempted to solve this problem by...
  15. I

    Chromatic Polynomial: Determine Pk(G) for Graph

    Homework Statement Determining the chromatic polynomial Pk(G) for following graph Homework Equations The Attempt at a Solution I did the solution in the picture so i get k and k-1 the colors should not be same as adjacent so that sounds like 2 colors is that all i need to...
  16. N

    Word problem polynomial f(x) help

    Homework Statement 2 People build a sandcastle of 36cm by 48cm by 60cm. Later they learn that the box will be smaller; a certain amount will be taken off each of the length width and height Homework Equations The Attempt at a Solution a) write a p(x) > (36-x) (48-x) (60-x)...
  17. E

    Taylor Polynomial Approximations.

    Hello, I'm new here, nice to meet you guys i was in class today and just didn't understand the taylor polynomial approximation, the professor started out approximating a function by polynomials of degree N, he first showed us how a linear polynomial was a crude approximation of the function...
  18. N

    Polynomial f(x) word problem help

    Homework Statement Kaylas lemonade stand has experienced rises /falls in sales caused by temperature changes over 2 summers. Her sales over the first two weeks of summer were tracked according to the model S(x) = x^3 -12x^2 +36x, where x is the number of days and S(x) is the number of sales...
  19. J

    Using Taylor Polynomial for Laplace Transforms

    Ive attached the problem and my work in the pic. Questions: Am I even applying the taylor polynomial the correct way? (I never learned taylor series, but I was supposed to be taught in the pre-requisite class) Am I suppose to plug in c=4? I am not so sure about how the U4(t) works...
  20. N

    Understanding Polynomial Functions: Analyzing h(x) = 3x + 2x

    Homework Statement Why is this not a polynomial function? h(x) = 3x + 2x Homework Equations The Attempt at a Solution 3x+2x = 5x. 5x is a linear function with a degree of 1, why is this not a polynomial funct?
  21. N

    Quick question of polynomial function

    Homework Statement Determien an euation for each polynomial function described below state whether the function is even, odd or neither. Sketch a graph of each. Homework Equations The Attempt at a Solution My question is.. If youre given information like this :: a) A quintic...
  22. N

    Polynomial function transformation help

    Homework Statement What is the difference between f(x)= x^3 , and g(x) = x^3 - x What does that -x do to the function and how do you represent it graphically? What about f(x) = 3x^3 +x Homework Equations The Attempt at a Solution No idea, wolframalphad the graph but i...
  23. N

    Determining if polynomial function

    Homework Statement We know that a polynomial function is anything with a positive exponent and a rational number. a) - / x^3 -4 d) 3x^-1 - 11 g) y= [sqrt of term]3x^2 -5x Homework Equations The Attempt at a Solution c) 1 / x^3 - 4 (do we use exponents to verify ?)...
  24. S

    Linear Algebra - polynomial functions of matrices

    Homework Statement Given p(x) = x4+2x2+1 and A = [[1 1 -2 0] [0 1 0 2] [1 1 -1 1] [0 0 -2 -1]] p(A) = 0 Find a polynomial q(x) so that q(A) = A-1 a) What is q(x)? b) Compute q(A) = A-1 Homework Equations I found the Cayley-Hamilton theorem, which states: p(x) = det(A-xIn)...
  25. X

    Perceived Ambiguity in Factoring Polynomial Expressions

    As it turns out, this first part was an extremely pervasive user error that I have not seen for days. Still, though, it is interesting that equations can be factored in many different ways, so I will post that here instead: 3x^3 = -5x^2 +2x -3x^3 -5x^2 + 2x = 0 Factored: x(-3x+1)(x+2) and...
  26. S

    Linear Algebra - Quadratic polynomial to Matrix

    Homework Statement Examining the answers of the previous two questions, write the quadratic polynomial f(x1,x2,x3)=x1x2−6x22+3x2x3−3x23 in the form f(x1,x2,x3)=[x1x2x3]A[x1x2x3]<-this last group is a column matrix where A is a symmetric matrix. Homework Equations Matrix multiplication...
  27. N

    What factors determine the exponent of a quartic polynomial function?

    Homework Statement (There is a comparison question I have, so i'll post my question with these) 8) Each of the following polonomials has x-intercepts of -6, 5 , and 0. Determien the appropriate equation for each. Then, sketch a graph of the function. b) A quartic function that extends...
  28. P

    Calculating Checksum: Frame Length vs Generator Polynomial

    To compute the checksum for some frame with m bits, corresponding to the polynomial M(x), the frame must be longer than the generator polynomial. Why...?
  29. N

    Principal Ideal, Polynomial generators

    Homework Statement Suppose R is an integral domain and I is a principal ideal in R[x], and I \neq {0} a) Show I = <g(x)> for some g(x)\inR[x] that has minimal degree among all non-zero polynomials in I. b) Is it necessarily true that I = <g(x)> for every g(x)\inR[x] that has minimal...
  30. P

    Taylor Polynomial Approximations (Apostol Section 7.8 #7)

    Homework Statement Prove that 0.493948<\int_0^{1/2} \frac{1}{1+x^4} dx<0.493958Homework Equations This chapter is about Taylor Polynomials, and specifically this section deals with Taylor's formula with remainder: f(x)=\sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k + E_n(x) The general formula for...
  31. D

    At what time do two particles meet: factoring a polynomial.

    Two particles move along the x-axis. Particle one has the position x = 8t^2 + 7t + 2 Particle two has the acceleration a = -8t, and when t=0 v=23. When the velocity of the particles match what is their velocity? I thought of approaching this problem by changing both equations into the...
  32. P

    Factoring a difficult polynomial so that I can extend equation

    Homework Statement Give a formula for the extended function that is continuous at the indicated point. (x3-4x2-11x+30)/(x2-4), x=2 Homework Equations The Attempt at a Solution I know that I have to factor the top and bottom so that I can cancel terms that cause the function to...
  33. P

    Trignometric Polynomial complex form

    Hi, I'm trying to learn Fourier transforms by myself. I'm a bit confused about how the trignometric polynomial complex form was derived. I'm referring to this: http://en.wikipedia.org/wiki/Trigonometric_polynomial Now, I haven't taken complex analysis so I only know the basics. I used Euler's...
  34. J

    Coefficients of characteristic polynomial

    I want to write an algorithm that gives as output the numbers a_n,\ldots, a_1,a_0, when a matrix A\in\mathbb{R}^{n\times n} is given as input, such that \det (A - \lambda) = a_n\lambda^n + \cdots + a_1\lambda + a_0,\quad\quad\forall\lambda\in\mathbb{C} If n=2, a_2 = 1,\quad a_1 =...
  35. P

    Dividing one polynomial by another

    \frac{r^3+3r^2+4r-8}{r-1} how do i solve that? and is there a general formula? thanks! Edit by bored Borek: LaTeX corrected.
  36. M

    What Are the Characteristic Polynomials of a Matrix Where A² = A?

    Homework Statement A n × n-matrix A satisfies the equation A2 = A. (a) List all possible characteristic polynomials of A. (b) Show that A is similar to a diagonal matrix Homework Equations The Attempt at a Solution A2 = A so, A2 - A = 0 A(A-I) = 0 Our minimal polynomial is x2 - x = m(x)...
  37. K

    Forcing a Least squares Polynomial through a fixed point

    Hi guys, Thanks for taking the time to read the post. I have a question related to curve fitting and polynomials that i was hoping someone might be able to help me with. I have a set of x and y data points, all on a graph. I have then calculated the 4th order least squares polynomial...
  38. Y

    Comparing coefficients of polynomial congruences.

    Today I read about the AKS - Primality test in which the simple theorem For gcd(a, n) = 1, we have (X - a)^n\equiv X^n - a\ (mod\ n)if and only if n is prime. was proven. The if direction is quite trivial from the fact that \binom{p}{k}\equiv 0\ (mod\ p) for 1\leq k < p. The other direction...
  39. C

    How Do You Calculate Taylor Polynomials for f(x,y) = ln(3y-8x) at Point (1,1)?

    Homework Statement f(x,y) = ln(3y-8x) Derive the first and second order Taylor polynomial approx, L(x,t) and Q(x,t), for T(x,T) about the point (1,1) Homework Equations -None- The Attempt at a Solution I do not understand what the question wants, nor do i want a solution. I...
  40. S

    Polynomial expression of Pendulum period with respect to angle (large)

    [b]1. Homework Statement The swinging period T(θ) for a small amplitude simple pendulum, is given by T, a constant for a constant length pendulum. If the initial angle θ is large, then the amplitude becomes large and the period needs to be corrected. The correction to the large amplitude...
  41. S

    Fitting a Second Order Polynomial to Data Points

    Homework Statement Suppose that you are given a set of observations (tk,yk), k = 1,...,M. You plot these points on a sheet & it seems that the relationship between (t,y) could be approximated with a second order polynomial. a) Write down the model in the form y = Ax + c. Specify the vectors...
  42. 8

    What Is the Minimal Polynomial of a=y^3 in the Field F=Kron(Z/2Z, x^4+x+1)?

    Homework Statement Find the minimal polynomial of a=y^3 in F=Kron(Z/2Z, x^4+x+1). (Calculate the powers of a^2, a^3, and a^4.) Homework Equations The Attempt at a Solution I attempted this trying to follow a similar worked problem in my book: a=y^3 & y^4=y+1 Multiply by...
  43. S

    How to Divide Polynomial Matrices Using Lambda Matrices

    Homework Statement Divide A(x)= [x3+2x2+3 -4x3-x2-5] [3x2-2 x3-2x2+x+4] by B(x) = [x+4 -3] [-x+6 x+2] on both the right side and the left side. Homework Equations The Attempt at a Solution I am...
  44. K

    Why must a polynomial be finite in length? Is there such thing as a

    Why must a polynomial be finite in length? Is there such thing as a recursive polynomial? Thanks guys!
  45. F

    Can all patterns in 3D have an equivalent equation in polynomial?

    Please Help me out with this one. Can all patterns in 2D or 3D, have an equivalent equation in polynomial?
  46. estro

    Characteristic Polynomial of A and A^2

    Hello, I'm trying to figure out connection between the characteristic polynomials for real matrices [3x3] and their powers. Suppose A is a real matrix [3x3] which's c.p is t^3+t^2+t-3, how can i find the c.p. of A^2. Now suppose p(t)=a_1t^3+a_2t^2+a_3t+a_4 Right away I can know that...
  47. P

    Linear recurrence with polynomial coefficients

    Hi all, I came across a linear recurrence with polynomial coefficients and realized that I don't have a clue as to how to solve it. The usual methods like generating functions or guessing seem not to work in that case. Here is the equation: i (i - 1) (i - 2) b[i] = 1/3 (i + 1) i (1 -...
  48. A

    Integration of exponential function times polynomial of fractional degree

    Homework Statement I'm working out a differential equation problem that I am supposed to solve with the formula \mathcal{L}\{t^\alpha\} = \frac{\Gamma{(\alpha + 1)}}{s^{\alpha+1}}. The problem is \mathcal{L}\{t^{\frac{1}{2}}\} (finding the Laplace transform of the given function)...
  49. D

    Dimension of a multivariate polynomial space

    Consider the space of all polynomials in n variables of degree at most d. The dimension of that space is C(n+d,d). How do I calculate the dimension of that same space when I restrict the domain of the polynomials to the unit ball? In that case all the polynomials (sum(i=1..n) x_i^2)^p with p a...
  50. C

    How do I show that a derivative of a polynomial has a zero in an interval?

    Homework Statement If an even degree polynomial of order 2n intersects the x-axis twice, how do I show that the (2n-1) th derivative has a zero in that interval? Homework Equations example: let g(x)=x^3(1-x). Show without computation that g'''(c) =0 for some c in (0, 1). The...
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