What is Polynomials: Definition and 783 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. G

    Classifcation of irreducible polynomials

    "Conjecture a classifucation rule for all irreducible polynomials of the form ax^2 + bx + c over the reals. Prove it." I'm stuck cold at the start. classification rule ? "Let R be an integral domain. A nonzero f in R[x] is irreducible provided f is not a unit and in every factorization f...
  2. C

    Finding Roots of Complex Polynomials: General Formula and Exponential Form

    Hi all Jut had a question. How do I go about finding the general formula for roots of the complex poly {z}^{n}-a where a is another complex number. Do I just go {z}^{n}=a? :S so complicated this things! Thanks in advance!
  3. T

    Find Legendre Polynomials of Order 15+

    Hey there, does anyone know where I could find a list of Legendre Polynomials? I need them of the order 15 and above, and I haven't been able to find them on the net. Thanks!
  4. S

    Splitting Polynomials Over Finite Fields: Fact or Fiction?

    Does anyone know if this is true and if so where they know it from? Given a polynomial over the integers there exists a finite field K of prime order p, such that p does not divide the first or last coefficient, and the polynomial splits over K. I realize this could be considered an...
  5. M

    Question regarding polynomials

    Hi This is the character equation for a polynomial of degree where n \geq 0 p(x) = a_0 x^{n} + a_{1} x^{n-1} + a_2 x ^{n-2} + \cdots + a_{n-1}x + a_{n} I'm presented with the following assignment: Two polynomials \mathrm{p, q} where n = 3. These polynomials can derived using the...
  6. C

    Resonance pde wave equation u(\phi,t) involving lagrange polynomials

    1/sin(phi) * d/d\phi(sin(phi) * du/d\phi) - d^2u/dt^2 = -sin 2t for 0<\phi < pi, 0<t<\inf Init. conditions: u(\phi,0) = 0 du(\phi,0)/dt = 0 for 0<\phi<pi How do I solve this problem and show if it exhibits resonance? the natural frequencies are w = w_n = sqrt(/\_n) =2...
  7. D

    Taylor differentition polynomials?

    taylor differentition polynomials? hi got a question here that involves this extremely difficult question anyone that can point me in the right direction on what to do will be most appreciated :) Find Exactly the tayor polynomial of degree 4 f(x) = cos ( pi*x / 6 ) about x=-1 i know...
  8. Z

    Is It Possible to Prove Normality of Polynomials?

    I was just wondering if it was possible to prove anything about the normality of the number: \sum_{x=0}^{\infty} \left((P(x) \mod b)\left(b^{-x}\right)\right) Where P(x) is a Polynomial with integer coefficients and b is the base of decimal representation. Is anything even known for...
  9. S

    Taylor Polynomial of 6th Degree for ln(1-x^2) with c=0

    I just want to check my answer. The question asks for the Taylor polynomial of degree 6 for ln(1-x^2) for -1<x<1 with c=0. I got tired after differentiating 6 times so I'm worried I made some mistakes along the way. The question also said: hint: evaluate the derivatives using the formula...
  10. T

    Proving Pn(x^2) as the 4n+2-nd Taylor Polynomial of sin(x^2) using Rn(x) Limits

    Show that Pn(x^2) is the 4n+2-nd Taylor polynomial of sin(x^2) by showing that \lim_{n\rightarrow infinity} R2n+1(x^2) = 0. note that Rn(x) represents the remainder I'm stuck on this question, can anyone help me please?
  11. S

    C/C++ Efficient Computation of Large Hermitian Polynomials in C++?

    Im having difficulty computing large Hermitian polynomials in C++. I fear I may have to steer away from a recursive formula. Any help would be greatly appreciated. John
  12. E

    Understanding Coordinate Vectors for Polynomials in P3

    Hi everyone, in this problem we are asked to get a coord vectors of polynomials with B as standard basis for P3 and then express one of the coord vectors as lin. combination of the others. So the set is this: {1-4x+4x^2+4x^3, 2-x+2x^2+x^3, -17 -2x-8x^2 + 2x^3} The way I was thinking is...
  13. G

    Which method is the most efficient for factoring polynomials?

    What is the fastest and easiest way to factor these? ex. 3x^2+8x-3
  14. W

    Solving quadratics and factorisation of polynomials using calculus

    I just found this really old book. In it, I found a way of solving quadratic equations using calculus. I've never seen this method in any other book. Ok, here's the method : The discriminant of the quadratic formula i.e sqrt(b^2 - 4ac) is equal to the first derivative of the original...
  15. D

    Maximum Number of Terms in a Homogeneous Polynomial of m Variables and Degree n

    I'm having a problem with a proof I came across in one of my calculus books but it's not the calculus part of the proof that I'm having trouble with. Here's the actual proof: "Prove: The number of distinct derivatives of order n is the the same as the number of terms in a homogeneous...
  16. H

    Understanding Asymptotes and Polynomials in Pre-Calculus

    Ok, I have a final in Pre-Calc comming up, and I am still a bit confused on finding asymtotes (vertical, horizontal, and slant) could someone help me with equations i can use to find the asymtotes or how i do? I am just really confused. Heres the problem. Find the vertical asymtote(s): F(x)...
  17. H

    Asymtotes and polynomials?

    Ok, I have a final in Pre-Calc comming up, and I am still a bit confused on finding asymtotes (vertical, horizontal, and slant) could someone help me with equations i can use to find the asymtotes or how i do? I am just really confused. Also, I am having trouble with finding a fourth degree...
  18. A

    Solving Polynomials (mod p) Problems

    I'm having problems finding all integer solutions to some of the higher degree polynomials. for p(x)= x^3− 3x^2+ 27 ≡ 0 (mod 1125), i get that 1125 = (3^2)(5^3). p(x) ≡ 0 (mod 3^2), p(x) ≡ 0 (mod 5^3). x ≡ 0, 3, 6 (mod 3^2) for 3^2 for 5^3, x ≡ 51 (mod 5^3) then i get x=801, 51, 426 (mod...
  19. A

    Simplify fractions of polynomials

    Simplify (x+1)/(x-1) multiplied by (x+3)/(1-x^2) divided by (x+3)^2/(1-x) Im not sure how to factor the 1-x^2 and what to do with 1-x I don't know how to simplify this please help someone. The answer to this question is 1/(x-1)(x+3) x cannot = 1,-1, and -3
  20. N

    Can someone explain this?(Taylor polynomials)

    For function f(x)=1/(1+x^2), calculate Taylor polynomials for the 2nd and 4th degree about the point a=0. The answer was: P2 = 1-x^2; P4 = 1-x^2+x.^4 for 2nd degree I got -2x/[(1+x^2)^2] for 4th degree I got 12x/[(1+x^2)^4]
  21. A

    Minimal Polynomials & Diagonalization: P_2(\mathbb{C}) & M_{k x k}(\mathbb{R})

    Compute the minimal polynomials for each of the following operators. Determine which of the following operators is diagonalizable. a) T : P_2(\mathbb{C}) \to P_2(\mathbb{C}), where: (Tf)(x) = -xf''(x) + (i + 1)f'(x) - 2if(x). b) Let V = M_{k \times k}(\mathbb{R}). T : V \to V[/itex]...
  22. H

    Root Theorem for Polynomials of Degree > 2

    What is the theorem that states if \Omega is a polynom with degree > 1 with real coefficients. If there exists a complex number z = a + bi such that \Omega(a+bi)=0 then \overline{z} = a - bi is also a root of \Omega ? For \Omega(x) = x^2 + px + q with p and q real then if a+bi is a...
  23. T

    A property of Chebyshev polynomials

    Hi, I fail finding a proof (even in MathWorld, in my Mathematic dictionary or on the Web) for the following property of Chebyshev polynomials: (T_i o T_j)(x) = (T_j o T_i)(x) = T_ij(x) when x is in ] -inf ; + inf [ Example : T_2(x) = 2x^2-1 T_3(x) = 4x^3-3x T_3(T_2(x)) = T_2(T_3(x)) =...
  24. T

    Legendre Polynomials: Beginner's Guide

    hi folks! I have been trying to figure out some plausible geometric intrepretation to legendre polynomials and what are they meant to do. I have come across the concept of orthogonal polynomials while working with some boundary value problems in solid mechanics and wasn't able to come to...
  25. V

    Evaluating the Product of Polynomials: (x-a)(x-b)(x-c)...

    well i could not get anything really mibd boggling, so u will have to put up with this one what is the product of: (x-a)(x-b)(x-c)..... = ?
  26. M

    Finding Polynomials with Specific Properties: How Do I Do That?

    Hi I got a Linear Algebra question. I'm suppose to find two polynomials p1 and p2 both of highest degree 3, and which satisfies the following: p1(-1) = 1 p1'(-1) = 0 p2(1) = 3 p2'(1) = 0 p1(0) = p2(0) p1'(0) = p2'(0) I hope that there is somebody out there...
  27. K

    Homework help (Basic Algebra-Division of polynomials)

    Please go to the bottom of this page for the problem that I am having trouble with.
  28. B

    Math Problem Inquiry Involving nonzero polynomials

    Find a nonzero polynomial f(w, x, y, z) in the four indeterminates w, x, y, and z of minimum degree such that switching any two indeterminates in the polynomial gives the same polynomial except that its sign is reversed. For example, f(z, x, y,w) = -f(w, x, y, z). Prove that the degree of the...
  29. D

    What's the purpose of Taylor Polynomials?

    I don't get it. I use it to approximate f for some x, but the formula for Taylor Polynomials already has f in it?
  30. I

    Lengendre polynomials

    I know that legendre polynomials are solutions of the differential equation is (1-x^2)d^2y/dx^2 - 2x dy/dx+l(l+1)y=0 where l is an integer. The first five solutions are P0(x)=1, P1(x)=x, P2(x)=3/2x^2-1/2, P3(x)=5/2x^3-3/2x, P4(x)=35/8x^4-15/4x^2+3/8 The problem is that I don't understand what...
  31. H

    Taylor Polynomials: Approximating f(x) and f'(x)

    Let f be a function that has derivatives of all orders for all real numbers. Assume f(1)=3, f'(1)=-2, f"(1)=2, and f'''(1)=4 a. Write the second-degree Taylor polynomial for f about x=1 and use it to approximate f(0.7). b. Write the third-degree Taylor polynomial for f about x=1 and use it...
  32. denian

    Factorising f(x) Completely: Find a & b Values

    given that f(x) = x^4 - 27x^2 - 14x + 120 can be expressed as ( x^2 + a )^2 - ( bx + 7 )^2 where a,b are constant. find the values of a and b. hence, or otherwise, factorise f(x) completely. the value of a and b are -13 and 1 respectively. so, f(x) = ( x^2 - 13 )^2 - ( x + 7...
  33. L

    How do I divide polynomials? (x^3-15x-7)/(x^2-3x-3)

    Ok, I have been trying to divide this polynomial. (x^3-15x-7)/(x^2-3x-3) After I factor the first part I get stuck. This is last problem on my homework and is due in less than an hour. Please some one help me out. Thanks
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