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. S

    Showing Difference of Relatively Prime Polynomials is Irreducible

    Homework Statement Let K be a field, and f,g are relatively prime in K[x]. Show that f-yg is irreducible in K(y)[x]. Homework Equations There exist polynomials a,b\in K[x] such that af+bg=u where u\in K. We also have the Euclidean algorithm for polynomials. The Attempt at a...
  2. S

    Vector Spaces, Polynomials Over Fields

    Vector Spaces, Polynomials "Over Fields" What does it mean when a vectors space is "over the field of complex numbers"? Does that mean that scalars used to multiply vectors within that vector space come from the set of complex numbers? If so, what does it mean when a polynomial, p(x) is...
  3. T

    Polynomials Problem Help: How to Solve P(z)Q(z)=0 Using Coefficient Equations

    Homework Statement [PLAIN]http://img443.imageshack.us/img443/3096/questiond.jpg The Attempt at a Solution If P(z)Q(z)=0 then \displaystyle a_0b_0 + (a_0b_1 + a_1 b_0)z + ... + \left( \sum_{i=0}^k a_i b_{k-i} \right) z^k + ... + a_n b_m z^{n+m} =0 Now what? Equate coefficients...
  4. C

    Integral involving Hermite polynomials

    Homework Statement The Hermite polynomials H_n(x) may be defined by the generating function e^{2hx-h^2} = \sum_{n=0}^{\infty}H_n(x)\frac{h^n}{n!} Evaluate \int^{\infty}_{-\infty} e^{-x^2/2}H_n(x) dx (this should be from -infinity to infinity, but for some reason the latex won't work!)...
  5. B

    Coprime Polynomials in K[X] and C[X]

    Let K be a subfield of C, the field of complex numbers, and f an irreducible polynomial in K[X]. Then f and Df are coprime so there exist a,b in K[X] such that af + bDf = 1 (D is the formal derivative operator). Now what I don't understand is why this equation implies f and Df are coprime when...
  6. C

    Determining a subspace of polynomials with degree 3

    Homework Statement Determine which of the following are subspaces of P3: a) all polynomials a0+a1x+a2x^2+a3x^3 where a0=0 b) all polynomials a0+a1x+a2x^2+a3x^3 where a0+a1+a2+a3=0 c) all polynomials a0+a1x+a2x^2+a3x^3 for which a0, a1, a2, a3 are integers d) all polynomials of the form...
  7. F

    Using Maclaurin Polynomials to Evaluate Trigonometric Functions at f(0.1)

    I am having some difficulty with a homework problem I was recently assigned. The problem says to "Replace each trigonometric function with its third Maclaurin polynomial and then evaluate the function at f(0.1)" This is what I have done so far: f(x)=(x cosx- sinx)/(x-sin⁡x) 1st trig...
  8. P

    Need a review on how to factor cubic polynomials

    So long story short, I have a friend who wants me to help her learn how to factor cubic polynomials. Normally I would just fess up and say I don't remember but it's something I'd like to review myself and lessons online aren't the clearest. Here's one of the questions: 2x3+3x2-8x+3 I...
  9. R

    Normalization constant for Legendre Polynomials

    Homework Statement I am following a derivation of Legendre Polynomials normalization constant. Homework Equations I_l = \int_{-1}^{1}(1-x^2)^l dx = \int_{-1}^{1}(1-x^2)(1-x^2)^{l-1}dx = I_{l-1} - \int_{-1}^{1}x^2(1-x^2)^{l-1}dx The author then gives that we get the following...
  10. N

    Factoring multivariable polynomials

    Hey, I'm a high school student (11th grade) and I'm working on a computer algebra system for a research project. Most things are are going well (sums, products, derivatives, integrals, series, expansion, complex analysis, factoring basic expressions, etc.). However, I am having difficulty...
  11. K

    What are some examples of irreducible polynomials in Z2[x]?

    (a) Find all irreducible polynomials of degree less than or equal to 3 in Z2[x]. (b) Show that f(x) = x4 + x + 1 is irreducible over Z2. (c) Factor g(x) = x5 + x + 1 into a product of irreducible polynomials in Z2[x]. We have an irreducible polynomial if it cannot be factored into a...
  12. T

    Solving Polynomials: Hints, Techniques & Solutions

    Homework Statement Solve for x, 225*sin(x)/x^6-225*cos(x)/x^5-90*sin(x)/x^4+15*cos(x)/x^3-5/(2*x^3)=0 Homework Equations Finding this very complicated to solve, are there any useful hints or techniques we should know about? The Attempt at a Solution Have used numerical...
  13. B

    Expanding 6x^2 in Terms of Legendre Polynomials

    Given the Legendre polynomials P0(x) = 1, P1(x) = x and P2(x) = (3x2 − 1)/2, expand the polynomial 6(x squared) in terms of P l (x). does anyone know what this question is asking me? what is P l (x)? thanks in advance
  14. W

    Anyone have any suggestions on books on chebyshev polynomials?

    i find that chebyshev polynomials are quite useful in numerical computations is there any good references?
  15. G

    Diff Eq's - orthogonal polynomials

    Diff Eq's -- orthogonal polynomials [PLAIN]http://img27.imageshack.us/img27/566/39985815.jpg I managed to do the first part, stuck in the part circled. Any help will be appreciated, thanks.
  16. R

    Taylor Polynomials- Lagrange remainder

    So I'm studying for a final, and it just so happens my professor threw taylor polynomials at us in the last week.. I understand the concept of a taylor polynomial but i need some help fully understand the LaGrange remainder theorem if we have a function that has n derivatives on the interval...
  17. A

    Can Bernstein Polynomials Approximate Functions and Their Derivatives Uniformly?

    Homework Statement Show that if f is continuously differentiable on [a, b], then there is a sequence of polynomials pn converging uniformly to f such that p'n converge uniformly to f' as well. Homework Equations The Attempt at a Solution Let pn(t) = cn t^n Use uniform convergence and integrate...
  18. K

    Division Algorithm For Polynomials

    Im given two polynomials: f(x) =(2x^6) + (x^5) - (3x^4) + (4x^3) + (x^2) -1 and g(x)=(x^3)-(x^2)+2x+3 find polynomials Q(x),R(x) in the set of R[x] s.t f(x) =g(x)Q(X) +R(X) and deg(R)<deg(g) Am i even in the right area? and something to do with manipulating numbers in C[x]...
  19. N

    Expand Polynomials: Finding Coefficients Using Pascal's Triangle

    Homework Statement Expand (a+b)n Homework Equations The Attempt at a Solution Substituting n=2, (a+b)n = a2 + 2ab + b2 Substituting n=3, (a+b)n = a3 + 3a2b + 3ab2 + b3 It's easy to see the powers of a decrease at the same time as the powers of b increase by order 1 each...
  20. C

    Complex Analysis (zeroes of Polynomials)

    I just wanted to know what kind of math is needed to solve questions like 1, 2 and 3 of http://www.math.toronto.edu/deljunco/354/ps4.fall10.pdf and number 5 of http://www.math.toronto.edu/deljunco/354/354final08.pdf . I don't need solutions, I just need to know what book or online source can...
  21. A

    Can the GCD of Polynomials in a Field Always Be Reduced to 1?

    Homework Statement We know that the gcd of two polynomials can be written as gcd(p(x),q(x))=p(x)m(x) + q(x)n(x) for some n(x) and m(x) in F[x] F a fieldI want to show gcd(n(x),m(x))=1 for a fixed gcd(p(x),q(x)) The Attempt at a Solution Well, what I tried was to let D(x)=gcd(p(x),q(x))...
  22. R

    Solving 4th Order Polynomials: Methods and Tips for Finding Roots

    Hey everyone Im doing control engineering and was wondering what methods i could use to find the roots of a 4th order polynomial? For example: (x^4) + (8x^3) + (7x^2) + 6x = 5 Could I separate that into two brackets of quadratics or will i need to use a really long winded method...
  23. P

    Linear Independence of Polynomials

    Homework Statement Given a set of polynomials in x: x^{r_1}, x^{r_2},...,x^{r_n} where r_i \neq r_j for all i \neq j (in other words, the powers are distinct), where the functions are defined on an interval (a,b) where 0 < a < x < b (specifically, x \neq 0), I'd like to show that this...
  24. C

    Algebra help - primitive roots and minimal polynomials

    Homework Statement (a) Find a primitive root β of F3[x]/(x^2 + 1). (b) Find the minimal polynomial p(x) of β in F3[x]. (c) Show that F3[x]/(x^2 + 1) is isomorphic to F3[x]/(p(x)). The Attempt at a Solution I am completely lost on this one :confused:
  25. L

    Taylor Polynomials: Find a0, a1, a2, a3, and a4

    Homework Statement Let f(x)=x2 +3x -5, and let the summation (from n=0 to infinity) an (x-4)n be the Taylor series of f about 4. Find the values of a0, a1, a2, a3, and a4. Homework Equations The Attempt at a Solution What am I supposed to do with the summation? And what does it mean...
  26. P

    Degrees of taylor polynomials

    Homework Statement use the third degree Taylor polynomial of cos at 0 to show that the solutions of x2=cos x are approx. \pm\sqrt{2/3}, and find bounds on the error. Homework Equations P2n,0(x) = 1-x2/2!+x4/4!+...+(-1)nx2n/(2n)! The Attempt at a Solution when it says "third...
  27. J

    How Do You Prove Equivalence of Two Polynomials?

    Can somebody prove the equivalence statement of two real polynomials in one variable x for me? My Math teacher just told us to remember it as a definition and so I didn't get any proof for it; I attempted to prove it myself and ended up confusing myself with a lot of symbols. So, can somebody...
  28. W

    Is Every Rootless Polynomial Over a Finite Field Prime?

    How to prove that a polynomial of degree 2 or 3 over a filed F is a prime polynomial if and only if the polynomial does not have a root in F? and i can't think of an example of polynomial of degree 4 over a field F that has no root in F but is not a prime polynomial. it says each...
  29. I

    Subspaces of polynomials with degree <= 2

    Homework Statement Which of these subsets of P2 are subspaces of P2? Find a basis for those that are subspaces. (Only one part) {p(t): p(0) = 2} Homework Equations The Attempt at a Solution So, I know the answer is that it's not a subspace via back of the book, but I don't...
  30. S

    Eigenvalue of vector space of polynomials

    Let V=C[x]10 be the fector space of polynomials over C of degree less than 10 and let D:V\rightarrowV be the linear map defined by D(f)=f' where f' denotes the derivatige. Show that D11=0 and deduce that 0 is the only eigenvalue of D. find a basis for the generalized eigenspaces V1(0), V2(0)...
  31. W

    What's the difference between polynomials and polynomial functions?

    What's the difference between polynomials (as elements of a ring of polynomials) and polynomial functions??
  32. silvermane

    Can Rational Functions be Written as a Sum of Polynomials?

    Homework Statement Take two polynomials f(x) and g(x) over a field, and suppose that gcd(f,g)=1. Show that any rational function, \frac{h(x)}{f(x)g(x)} can be written in the form (\frac{p(x)}{f(x)}) + (\frac{q(x)}{g(x)}) for some polynomials, p and q. The Attempt at a Solution I...
  33. C

    Dimension of subspace of even and odd polynomials

    Homework Statement I have a question which asks me to find the dimensions of the subspace of even polynomials (i.e. polynomials with even exponents) and odd polynomials. I know that dim of Pn (polynomials with n degrees) is n+1. But how do I find the dimensions of even n odd polynomials...
  34. A

    Show that the Hermite polynomials H2(x) and H3(x).

    Hi guys. I am new, and i need help badly. I have been asked this question and I have no idea how to do it. Any help would be appreciated! Show that the Hermite polynomials H2(x) and H3(x) are orthogonal on x € [-L, L], where L > 0 is a constant, H2(x) = 4x² - 2 and H3(x) = 8x³ - 12x...
  35. silvermane

    Polynomials and Calculating via mod7

    Homework Statement Working in (Z/7Z)[x], compute the greatest common divisor of the polynomials (X^2 - 3X + 2) and (X+6). The Attempt at a Solution I need help understanding how to compute this. It would be greatly appreciated if someone gave me an example with different modulus and...
  36. silvermane

    How to Determine Irreducible Polynomials in (Z/2Z)[x]?

    Homework Statement Find all monic polynomials of degrees 2 and 3 in (Z/2Z)[x]. Determine which ones are irreducible, and write the others as products of irreducible factors. The Attempt at a Solution I know that factors of degree 1 correspond to roots in Z/2Z and that monic...
  37. L

    Interpolation using Lagrange polynomials

    Problem: We want to calculate a polynomial of degree N-1 that crosses N known points in the plane. Solution A: solving a NxN system of linear equation (Gauss elimination) Solution B: construction from Lagrange basis polynomials. One of my professors said that the first solution is...
  38. P

    Legendre Polynomials - expansion of an isotropic function on a sphere

    Hello. I don't know what to do with one integral. I am sure it is something very simple, but I just don't see it... For some reason I am not able to post the equations, so I am attaching them as a separatre file. Many thanks for help.
  39. V

    Prime values of integer polynomials

    Hey there, physics forums! A question occurred to me the other day: Is it true that if f \in \mathbb{Z}[x] is monic and irreducible over \mathbb{Q} , then for at least one a \in \mathbb{Z} , f(a) is prime? I can't prove it, but I suspect it's true. Does anyone know if this problem...
  40. O

    Rewriting polynomials for computers

    Suppose I have a REALLY big polynomial: a_0 + a_1 x + a_2 x^2 + a_3 x^3+a_4 x^4+ \cdots + a_n x^n I can rewrite the polynomial as a combination of multiplication and addition operators (instead of exponents) that a computer tends to like as such: a_0 + x \left( a_1 + x \left( a_2 + x \left(...
  41. C

    Finding kernel and range for polynomials transformation

    I have troubles arriving at the solution to this question: Consider the transformation T: P3-->P3 given by: T(f)=(1-x^2)f '' - 2xf ' Determine the bases for its range and kernel and nullity and rank Can anyone explain how should i go about finding the bases for its kernel and range?? i get 0...
  42. M

    Proving the Limit of f(x)-g(x) for Even Integer Polynomials

    Homework Statement I want to prove that a polynomial f(x) and a polynomial g(x) with degrees of k,n where k,n are positive even integer, n>k that limit x-> - infinity of f(x)-g(x)=-infinity Homework Equations a polynomial can be written as a1x^n+a2x^(n-1)...+a(n-1)x+an The...
  43. X

    Orthogonality of Legendre Polynomials

    Homework Statement For spherical coordinates, we will need to use Legendre Polynomials, a.Sketch graphs of the first 3 – P0(x), P1(x), and P2(x). b.Evaluate the orthogonality relationship (eq 3.68) to show these 3 functions are orthogonal to each other. (3 integrals). c.Show that the...
  44. K

    General Formula for Multiplying Polynomials?

    Homework Statement Does a general formula exist? \sum \limits_{k=0}^{m_1} a_kx^k\cdot\sum \limits_{k=0}^{m_2} b_kx^k=\sum \limits_{k=0}^{m_1+m_2} c_kx^k Homework Equations The Attempt at a Solution I am having trouble understanding the relation between the c coefficients in the product and...
  45. R

    Solving Polynomial Roots: Sum of Cubes and Fourth Powers - Further Maths Help

    Show that the sum of the cubes of the root of the equation x3 + (lambda)x + 1 = 0 is -3 Show also that there is no real value of (lambda) for which the sum of the fourth powers of the roots is negative. This question is in one of the past papers of further maths and I don't know how...
  46. T

    How Do You Solve Complex Fraction Equations?

    Homework Statement Solve for X. Homework Equations (3 / x+2) - (1 / x) = 1 / 5x The Attempt at a Solution (NOTE: I always had difficulties with fractions) (3x - x - 2 / x² + 2) - (1 / 5x) = 0 (2x - 2 - x² - 2) / 5x³ + 10x (-x² + 2x - 4) / 5x³ + 10x ^ I attempted a few...
  47. L

    Finite fields and products of polynomials

    Homework Statement This question is in two parts and is about the field F with q = p^n for some prime p. 1) Prove that the product of all monic polynomials of degree m in F is equal to \prod (x^(q^n)-x^(q^i), where the product is taken from i=0 to i=m-1 2) Prove that the least common multiple...
  48. U

    A level Further Pure Maths help (Polynomials)

    Homework Statement Find the values of Σ(a^2), Σ(1/a), Σ(a^2)(B^2) and ΣaB(a + B) for: x^4 - x^3 + 2x + 3 = 0 Homework Equations Σa = 1, ΣaB = 0, ΣaBC = -2, aBCD = 3 The Attempt at a Solution I found the Σ(a^2) and Σ(1/a) successfully correct bt could neither find Σ(a^2)(B^2)...
  49. K

    Solving Polynomials - Answers to Common Questions

    i'm pretty new to polynomials and I have qns, which I do not know how to solve. x5 + ax3 + bx2 - 3 = (x2 - 1)Q(x)- x - 2 Q(x) is a polynomial. Solve a and b. I know the degree of Q(x) is 3, so I subst Q(x) into x3, but I got stuck. Help. Thanks in advance.
  50. E

    Strategies for constructing Maclaurin polynomials?

    I came across a problem in my homework to construct a MacLaurin polynomial of the nth degree for \sqrt{1+x}, and had some major problems. I gave up and looked up the answer on the internet, which was fairly complex: \sum \frac{(-1)^{n}(2n)!x^{n}}{(1-2n)(n!)^{2}(4^{n})} Well, I know I...
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