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.

View More On Wikipedia.org
Recent contents Top users
  1. greg_rack

    B Super silly question about a polynomial identity

    Am I(always) legitimized to write ##-(a-b)^n=(b-a)^n##? I don't know why but it's confusing me... can't really understand when and why I can use that identity
  2. anemone

    MHB Polynomial with integer coefficients

    Let $a,\,b,\,c$ be three distinct integers and $P$ be a polynomial with integer coefficients. Show that in this case the conditions $P(a)=b,\,P(b)=c,\,P(c)=a$ cannot be satisfied simultaneously.
  3. Marso

    Multiple Questions — Evaluating the graph of this polynomial

  4. Y

    How Do I Solve Polynomial Problem #19?

    I have problem solving this. The question is #19 in the first attachment. My work is in the second attachment I work to this point and get stuck: https://www.physicsforums.com/attachments/270843 Can anyone help me? Thanks
  5. anemone

    MHB First, second and third derivatives of a polynomial

    Let $p(x)$ be a polynomial with real coefficients. Prove that if $p(x)-p'(x)-p''(x)+p'''(x)\ge 0$ for every real $x$, then $p(x)\ge 0$ for every real $x$.
  6. S

    Remainder of polynomial division

    ##x^{2017} + 1 = Q(x) . (x-1)^2 + ax + b## where ##Q(x)## is the quotient and ##ax+b## is the remainder ##x=1 \rightarrow 2 =a+b## Then how to proceed? Thanks
  7. K

    I Finite fields, irreducible polynomial and minimal polynomial theorem

    I thought i understood the theorem below: i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field Then this example came up: The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...
  8. J

    Legendre polynomial - recurrence relations

    Note: $P_n (x)$ is legendre polynomial $$P_{n+1}(x) = (2n+1)P_n(x) + P'_{n-1}(x) $$ $$\implies P_{n+1}(x) = (2n+1)P_n(x) + \sum_{k=0}^{\lfloor\frac{n}{2}\rfloor} (2(n-1-2k)+1)P_{n-1-2k}(x))$$ How can I continue to use induction to prove this? Help appreciated.
  9. anemone

    MHB Prove No Integers Solve $ax^3+bx^2+cx+d=1$ for x=19,2 for x=62

    Prove that there are no integers $a,\,b,\,c$ and $d$ such that the polynomial $ax^3+bx^2+cx+d$ equals 1 at $x=19$ and 2 at $x=62$.
  10. anemone

    MHB Area of the bounded regions between a straight line and a polynomial

    Let $P$ be a real polynomial of degree five. Assume that the graph of $P$ has three inflection points lying on a straight line. Calculate the ratios of the areas of the bounded regions between this line and the graph of the polynomial $P$.
  11. S

    What is the Remainder When f(x) is Divided by (x+1)?

    ##f(x)## is divisible by ##(x-1) \rightarrow f(1) = 0## ##f(x) = Q(x).(x-1)(x+1) + R(x)## where ##Q(x)## is the quotient and ##R(x)## is the remainderSeeing all the options have ##f(-1)##, I tried to find ##f(-1)##: ##f(-1) = R(-1)## I do not know how to continue Thanks
  12. anemone

    MHB Finding $|k|$ of the Polynomial $x^3-kx+25$

    The polynomial $x^3-kx+25$ has three real roots. Two of these root sum to 5. What is $|k|$?
  13. anemone

    MHB Polynomial Challenge: Show Real Roots >1 Exist

    If the equation $ax^2+(c-b)x+e-d=0$ has real roots greater than 1, show that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.
  14. anemone

    MHB Evaluate the constant in polynomial function

    Let $a,\,b,\,c,\,d,\,e,\,f$ be real numbers such that the polynomial $P(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f$ factorizes into eight linear factors $x-x_i$ with $x_i>0$ for $i=1,\,2,\,\cdots,\,8$. Determine all possible values of $f$.
  15. anemone

    MHB Solving a Third-Degree Polynomial with Real Coefficients

    Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12$. Find $|f(0)|$.
  16. G

    How to expand this ratio of polynomials?

    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...
  17. D

    MHB Vector of polynomial and basis.

    If this question is in the wrong forum please let me know where to go. For p, the vector space of polynomials to the form ax'2+bx+c. p(x), q(x)=p(-1) 1(-1)+p(0), q(0)+p(1) q(1), Assume that this is an inner product. Let W be the subspace spanned by . a) Describe the elements of b) Give a basis...
  18. chwala

    Finding this constant in a quartic polynomial

    ##x^2(3x^2+4x-12) +k=0## ##(3x^2+4x-12)= \frac{-k}{x^2}## or ##(4x^3-12x^2)=-k-3x^4## ##4(3x^2-x^3)=3x^4+k## ##4x^2(3-x)= 3x^4+k## or using turning points, let ##f(x)= 3x^4+4x^3-12x^2+k## it follows that, ##f'(x)=12x^3+12x^2-24x=0## ##12x(x^2+x-2)=0## ##12x(x-1)(x+2)=0## the turning points...
  19. anemone

    MHB Real Roots of Polynomial Minimization Problem

    For an integer $n\ge 2$, find all real numbers $x$ for which the polynomial $f(x)=(x-1)^4+(x-2)^4+\cdots+(x-n)^4$ takes its minimum value.
  20. anemone

    MHB Roots of a Polynomial Function A²+B²+18C>0

    If a polynomial $P(x)=x^3+Ax^2+Bx+C$ has three real roots at least two of which are distinct, prove that $A^2+B^2+18C>0$.
  21. Zack K

    Infinite Square Well with polynomial wave function

    Some questions: Why is this even a valid wave function? I thought that a wave function had to approach zero as x goes to +/- infinity in all of space. Unless all of space just means the bounds of the square well. Since we have no complex components. I am guessing that the ##\psi *=\psi##. If...
  22. F

    Polynomial ring zero divisors

    Proof: ##(\Leftarrow)## Suppose there exists non zero ##b \in R## such that ##bp(x) = 0##. Well, ##R \subset R[x]##, and so by definition of zero divisor, ##p(x)## is a zero divisor. (assuming ##p(x) \neq 0##). ##(\Rightarrow)## Suppose ##p(x)## is a zero divisor in ##R[x]##. Then we can choose...
  23. Lauren1234

    Possible webpage title: Can You Solve the No Snap Order Puzzle with Pearls?

    This is my solution however I feel like the number is far too big can anyone see what I’ve done wrong
  24. C

    How can I evaluate a Chebishev polynomial in python?

    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...
  25. H

    I Polynomial approximation of a more complicated function

    There is an arbitrarily complicated function F(x,y,z). I want to find a simpler surface function G(x,y,z) which approximates F(x,y,z) within a region close to the point (x0,y0,z0). Can I write a second-order accurate equation for G if I know F(x0,y0,z0) and can compute the derivatives at the...
  26. T

    MHB Solving a Quartic Polynomial with Symmetric Graph & Intercept -2

    Find the equation of a quartic polynomial whose graph is symmetric about the y -axis and has local maxima at (−2,0) and (2,0) and a y -intercept of -2
  27. Monoxdifly

    MHB [ASK] Determinant of a Matrix with Polynomial Elements

    Help me if what I have done so far can be simplified further.
  28. S

    Python Finding the coefficients of a Taylor polynomial

    To find the coefficients of the Taylor polynomial of degree two of the function ##z(x,y)## around the point ##(0,0)##, what would be a handy way of doing that in python? How would one find the derivatives of ##z(x,y)##?
  29. A

    MHB Rescaled Coordinates in a polynomial equation.

    I have this question from Murdock's textbook called: "Perturbations: Methods and Theory": Use rescaling to solve: $\phi(x,\epsilon) = \epsilon x^2 + x+1 = 0$ and $\varphi (x,\epsilon) = \epsilon x^3+ x^2 - 4=0$. I'll write my attempt at solving these two equations, first the first polynomial...
  30. S

    Lill's method for solving polynomial equations

    Summary: Worth teaching in secondary school? - or too bewildering? The mathologer video made me aware of Lill's method for solving polynomials with real roots. Although I'm not involved in secondary school teaching, I can't help wondering if it is a suitable topic for that level. Perhaps...
  31. S

    Solve polynomial using complex number

    I can do question (a). For question (b), I can not see the relation to question (a). Can we really do question (b) using result from (a)? Please give me little hint to relate them Thanks
  32. Adgorn

    Limit of the remainder of Taylor polynomial of composite functions

    Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...
  33. C

    Finding values for a and b for this polynomial

    the polynomial f(x) = ax4 - 3x3- 63x2+ 152x - b has one of its zeros at x = 5 and passes through the point (-2, -560) Question: Use this info to find the values of a and b I am prepping for a test and this one question is really stumping me, I wondered if anyone would be able to help. For all...
  34. C

    MHB Zero divisor for polynomial rings

    Dear Everybody,I am having trouble with how to begin with this problem from Abstract Algebra by Dummit and Foote (2nd ed): Let $R$ be a commutative ring with 1. Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a zero divisor in...
  35. N

    MHB How can I determine a polynomial is irreducible in C？

    Hello!(Blush) I learned about Eisenstein's irreducibility criterion. But it's still hard for me to implement it when the integer coefficients may be as larger as 1e9. What's more, how can I (or my computer?) know when to change x into x+a? It really puzzles me(O_o)??
  36. R

    B Properties of roots of polynomials

    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...
  37. R

    B Roots of Polynomials: Understanding Mathematical Methods

    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...
  38. hilbert2

    Complex polynomial on the unit circle

    So, the values of polynomial ##p## on the complex unit circle can be written as ##\displaystyle p(e^{i\theta}) = a_0 + a_1 e^{i\theta} + a_2 e^{2i\theta} + \dots + a_n e^{ni\theta}##. (*) If I also write ##\displaystyle a_k = |a_k |e^{i\theta_k}##, then the complex phases of the RHS terms of...
  39. SamRoss

    B Trouble with polynomial long division

    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?
  40. Leo Consoli

    Find the number of integer solutions of a second degree polynomial equation

    x^2 - x -3 + 2c = 2x(ax+b) x^2 -2ax^2 - 2bx - x - 3 + 2c = 0 x^2(1-2a) -x(1+2b) -3 + 2c =0 Using girard r1+r2 = (1+ 2b)/(1-2a) r1xr2 = (-3 +2c)/(1-2a) After this I am stuck. Thank you.
  41. M

    Integral question on a polynomial

    At first I was thinking about using the dirac delta function ##\delta(x-1)##, but then I recalled ##\delta \notin L_2[0,1]##. Any ideas? I'm thinking no such function exists.
  42. S

    Remainder of a polynomial

    f(x) = A(x) . (x2 + 4) + 2x + 1 f(x) = B(x). (x2 + 6) + 6x - 1 f(x) = C(x) . (x2 + 6) . (x2 + 4) + s(x) Then I am stuck. What will be the next step? Thanks
  43. M

    MATLAB Solving Polynomial Eigenvalue Problem

    Hi PF! I'm trying to solve the polynomial eigenvalue problem ##M \lambda^2 + \Phi \lambda + K## such that K = [5.92 -.9837;-0.3381 109.94]; I*[14.3 24.04;24.04 40.4]; M = [1 0;0 1]; [f lambda cond] = polyeig(M,Phi,K) I verify the output of the first eigenvalue via (M*lambda(1)^2 +...
  44. Mr Davis 97

    I Polynomial of finite degree actually infinite degree?

    ##1+x+x^2 = \dfrac{1-x^3}{1-x} = (1-x^3)\cdot \dfrac{1}{1-x} = (1-x^3)\sum_{k=0}^\infty x^k##. Isn't this a contradiction since the LHS has degree ##2## while the RHS has infinite degree?
  45. M

    MHB Least squares method : approximation of a cubic polynomial

    Hey! :o I want to determine an approximation of a cubic polynomial that has at the points $$x_0=-2, \ x_1=-1, \ x_2=0 , \ x_3=3, \ x_4=3.5$$ the values $$y_0=-33, \ y_1=-20, \ y_2=-20.1, \ y_3=-4.3 , \ y_4=32.5$$ using the least squares method. So we are looking for a cubic polynomial $p(x)$...
  46. V

    Matlab Plotting Points and a Cubic Polynomial that passes through them

    Homework Statement We were given a tutorial to complete which I did complete. Now the question is: By modifying the appropriate lines in your script file, find the values of a, b, c, and d so that the cubic polynomial  y = ax3 + bx2 + cx + d  passes through the (x, y) pairs (-1, 3), (0, 8)...
  47. R

    Given a polynomial p, prove A^2-B^2 = p(1)p(-1)

    Homework Statement Given a polynomial p, let A be the sum of the coefficients of the even powers, and let B be the sum of the coefficients of the odd powers. Prove that A^2 - B^2 = p(1)p(-1). Homework EquationsThe Attempt at a Solution See attached. Can someone please look at my work to see if...
  48. P

    MHB Understand Polynomial Terms: Like & Unlike Terms

    I would like to have someone who would be willing to explain me what is a like term and an unlike term in terms of set theory. I'm just an high-scool student, but I really would like to understand it from that point of view anyway. It doesn't have to be a 1000 pages long of explanations.
  49. Adgorn

    B Questions regarding polynomial divisions and their roots

    Hello everyone, Going through calculus study, there is a vague point regarding polynomials I'd like to make clear. Say there's a polynomial ##f## with a root at ##a## with multiplicity ##2##, i.e. ##f(x)=(x-a)^2g(x)## where ##g## is some other polynomial. I define ##h(x)=\frac {f(x)} {x-a}##...
  50. YoungPhysicist

    Find the function that matches the equation

    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...
Back
Top