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. C

    Determining the second order polynomial from the intersection points

    Homework Statement Let's say that we have a second order polynomial function, and we know all of the points where it intersects with the x and y axis. Ex: (-2; 0), (0; 2), (1; 0) How does on determine the ax^2+bx+c polynomial form based on that? Homework Equations - The Attempt at...
  2. H

    Writing out the span of this polynomial vector space?

    Homework Statement The problems states "All polynomials of the form p(t)= at^2, where a is in R." I'm supposed to see if it is a subspace of Pn. I've already done that but the book's answer is that it spans Pn by Theorem 1, because the set is span{t^2} Homework Equations Theorem states "1 If...
  3. U

    Solving a Polynomial with Real Coefficients and Real Zeroes

    Homework Statement Let f(x) = x^{4}+ax^{3}+bx^{2}+cx+d be a polynomial with real coefficients and real zeroes. If |f(i)| = 1, (where i = \sqrt{-1}) then find a+b+c+d. Homework Equations The Attempt at a Solution f(i) = 1-b+d+ci-ai Taking modulus |f(i)|= |1-b+d+i(c-a)|...
  4. A

    How do you find the minimal polynomial?

    Homework Statement If we have a transformation matrix \begin{bmatrix} 1 & 2 & 4 \\0 & 0 & 0 \\0 & 0 & 0 \end{bmatrix} Homework Equations The Attempt at a Solution I found the characteristic polynomial of this matrix: x^3 - x^2 = x^2(x-1) ...can anybody please help me...
  5. N

    Finding polynomial function with given zeros

    Homework Statement find the polynomial function p(x) with zeros, -1, 1, 3 and P(0)=9 Homework Equations all i have is (x^2-1) and (x-3) The Attempt at a Solution
  6. N

    Is 2 a Root of the Polynomial 4x^3 - 3x^2 - kx - 4k^2 = 0?

    Homework Statement if 2 is a root of 4x^3-3x^2-kx-4k^2= 0 find th value of k Homework Equations The Attempt at a Solution
  7. J

    Showing a polynomial has at least one zero outside the unit circle.

    The first thing that we should notice is that the leading coefficient $a_n = 1$. I was thinking about considering the factored form of p. I googled, and there is an algorithm called the "Schur-Cohn Algorithm" that is suppose to answer exactly this, but I can't find any information on it or...
  8. 1

    Polynomial Span related problem Linear Algebra

    Homework Statement Consider the vector space F(R) = {f | f : R → R}, with the standard operations. Recall that the zero of F(R) is the function that has the value 0 for all x ∈ R: Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have the same value at x = −1 and x = 1...
  9. 1

    Polynomial Span and Subspace - Linear Algebra

    Homework Statement Consider the vector space F(R) = {f | f : R → R}, with the standard operations. Recall that the zero of F(R) is the function that has the value 0 for all x ∈ R: Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have the same value at x = −1 and x = 1...
  10. M

    Taking the derivative of a polynomial fraction?

    Taking the derivative of a polynomial fraction?? b]1. Homework Statement [/b] Ok, so the question wants me to differentiate f(x)= (x)/(x+1). We are supposed to use the definition of the derivative f'(x)= (limit as h->0) [f(x+h)-f(x)]/(h). We also have learned the power rule. I did the formula...
  11. T

    Quick check of 4 term polynomial factorised

    Could someone quickly go over my working, as I am not 100% sure I have done it the right way. I will show and explain my working step by step. $$ at^2-4a + 2t^2-8$$ I first grouped the values: (at^2-4a) + (2t^2-8) I then factorised these equations into: a(t^2-4a) + 2(t^2-4) I...
  12. B

    MHB Proving an entire function is a polynomial under certain conditions

    Hello, This was an exam question which I wasn't sure how to solve: Suppose f is entire and |f(z)| \leq C(1+ |z|)^n for all z \in \mathbb{C} and for some n \in \mathbb{N}. Prove that f is a polynomial of degree less than or equal to n. I know that f can be expressed as a power series, but I'm...
  13. tom.stoer

    Solution of polynomial equations

    Suppose there is a set of complex variables \{x_i,\,i=1 \ldots M;\;\;y_k,\,k=1 \ldots N\} and a polynomial equation p(x_i, y_k) = 0 Is there a way to prove or disprove for such an equation whether it can be reformulated as f(x_i) = g(y_k) with two functions f and g with...
  14. ElijahRockers

    Determining Polynomial Subspaces in P4

    Homework Statement Determine whether the following are subspaces of P4: a) The set of polynomials in P4 of even degree b) The set of all polynomials of degree 3 c) The set of all polynomials p(x) in P4 such that p(0) = 0 d) The set of all polynomials in P4 having at least one real root The...
  15. J

    Where g is a polynomial function of degree n-1

    Let [ tex ]f(x)=\sum_{i=0}^n c_i x^i[ / tex ] be an arbitrary polynomial function of degree n Show that if f(0)=0 then either f is constant or f(x)=xg(x), where g is a polynomial function of degree n-1 I don't know how to start. Please help Thank you in advance
  16. S

    A question on the orthogonal polynomial

    Dear All Friends, I am currently working on a project which needs some orthogonality integration formulae of Laguerre polynomials. I referred worlfram's math function site http://functions.wolfram.com/Polynomials/LaguerreL3/21/02/01/ and get three seemingly useful ones...
  17. R

    Unique factorization domain, roots of a polynomial, abstract algebra

    Homework Statement let A be a UFD and K its field of fractions. and f\in A[x] where f(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} is a monic polynomial. Prove that if f has a root \alpha=\frac{c}{d}\in K,K=Frac(A) then in fact \alpha\in A I need some guidance with the proof. Proof...
  18. T

    If the roots of a polynomial p are real, then the roots of p' are real.

    Homework Statement Let p be a polynomial. Show that the roots of p' are real if the roots of p are real. Homework Equations The Attempt at a Solution So we start with a root of p', call it r. We want to show that r is real. Judging by the condition given, I am assuming that...
  19. K

    Irreducible polynomial in Q[x] but redicuble in Z[x]

    Homework Statement Give an example of a polynomial irreducible in Q[x], but reducible in Z[x] Homework Equations The Attempt at a Solution I think there is no example of this. The coefficients of a reducible polynomial with integer coefficients will be rational, or am I mistaken?
  20. I

    Prove AB and BA have the same characteristic polynomial

    Homework Statement Show that for any square matrices of the same size, A, B, that AB and BA have the same characteristic polynomial. Homework Equations The Attempt at a Solution I understand how to do this if either A or B is invertible, since they would be similar then. I saw a...
  21. dexterdev

    How Does a Minimal Polynomial Differ from a Characteristic Polynomial?

    I have a small idea on what irreducible and primitive polynomials are in Abstract algebra. But what is minimal polynomial? -Devanand T
  22. M

    Integrating a complicated polynomial

    Homework Statement Hi An integration question: t= 1/(1+r2) Can you please show me how to integrate with respect to r. Thank you. The Attempt at a Solution ∴t = (1+r2)-1 I then tried substituting u = 1+r^2 but that didn't work! Is there a trick with this?
  23. J

    Computation Question in the Ring of Polynomial with Integer Coefficients

    I have a quick question. The problem reads: Prove that there is no integer m such that 3x2+4x + m is a factor of 6x4+50 in Z[x]. Now, Z[x] is not a field. So, the division algorithm for polynomials does not guarantee us a quotient and remainder. When I tried dividing 6x4+50 by 3x2+4x +...
  24. S

    Polynomial Remainder Therem to proove this

    Homework Statement Applying remainder theorem again and again to show that the remainder of the f(x) polynomial function when divided by (x-α)(x-β) is A(x-α)+B . Determine A and B Homework Equations the remainder of a polynomial f(x), divided by a linear divisor x-a, is equal to f(a) The...
  25. A

    How to make 4 variable polynomial equatio

    Can I use excel to make an equation for 4 variables (x,y,z,w) e.g a,b,c,d,3,f,g,h,i, would be constants w = ax + by + cz + dx^2 + ey^2 + fz^2 + gxy + hyz + iyz What are these equations called ? What literature can I study to better understand the procedure of making these tye of...
  26. A

    Orthogonal properties of associated laguerre polynomial

    i need the derivation of orthogonal properties of associated laguerre polynomial (with intermediate steps). someone please tell me where can i get it (for easy understanding).
  27. marellasunny

    The Degree of the Zero Polynomial: Why is it Defined as -∞?

    I understand that mathematicians have had to define the number '0' also as a polynomial because it acts as the additive identity for the additive group of poly's.What I do not understand is why they define the degree of the zero polynomial as [ tex ]-\infty[ /tex ]. An explanation on planetMath...
  28. D

    Calculate a polynomial function from other polynomial functions

    Background information: I have come up against a mathematical question which I as somebody with relatively limited exposure to maths can not seem to answer. I am a student working on a thesis dealing with near-infrared spectroscopy (NIRS). The NIR scanner is able to measure the moisture content...
  29. M

    Finding number of zeroes in a polynomial?

    Let's say I have the equation f(x) = 2x + 3 * (3x^2 + 3) - x^2 + 5. If my algebra is right, this is a 3rd-degree polynomial. How many zeroes does this equation have? How did you figure that out?
  30. Hercuflea

    Polynomial higher order DE in D notation form.

    Homework Statement My teacher likes to teach us the D-notation methods for higher order DE's. I am having a hard time with this one and I can't seem to find the formula for the general solution Find a fundamental set of the equation (D-1)^{2}(D^{2}-6D+13)^{3}y = 0 Homework...
  31. K

    How can I factor a 3rd order polynomial using the cubic formula?

    Homework Statement What are the steps to factoring 3rd order polynomials like x^3+8x^2-21x+10? It's to find eigenvalues of a matrix in linear algebra, I completely forgot how to factor and it's killing me. Homework Equations The Attempt at a Solution None, unless its a polynomial...
  32. W

    What would happen if you try to fit 'n' degree polynomial to (n+1) data points?

    I couldn't get these lines from my book. I will reproduce it here. Warning: In step 1, if you use computer to fit a polynomial to the data , it could lead to disaster. For example, consider fitting a sixth degree polynomial to the seven data points, or, an (n-1) degree polynomial to n...
  33. P

    MHB SE Class 10 Maths - Factor theorem and its applications

    factor completely 2x3 - 8a2x + 24x2 + 72x so far i got 2x(x2 + 12x + 36) - 8a2x 2x(x+6)(x+6) - 8a2x don't know what to do from here.
  34. M

    Calculators TI-83/84 Plus - Storing polynomial functions

    Unless I'm missing something here, I've noticed that if you want to store a polynomial function on the TI-83 or the TI-84 Plus, you have to create a program that asks you what the value of x is, then displays the value of f(x). I kind of wish I could define a function without making a program.
  35. M

    How to find polynomial roots on a TI-83 or TI-84 Plus without PolySmlt?

    I'd like to know how to find the root of a polynomial on my TI-84 Plus without this "Polynomial Root Finder and Simultaneous Equation Solver" app. The reason is that the app's not in my calculator and I can't transfer the app to my calculator. I keep getting an "Access Denied" error message...
  36. P

    Ratio of two polynomial functions - integral

    The problem is :- Integral of (1+x^4) / ( 1 + x^6) . dx I have reduced it to a form of Integral of 2.sqrt(tantheta) / ( 1 + tan^3 theta ) over dtheta where x^2 = tantheta. However I cannot reduce it further. How do I proceed ? In general, how do I proceed given a problem of the form...
  37. V

    What Exactly are the Roots of a Polynomial?

    Are the roots of a polynomial given by the function f(x) defined as the values for x where f(x)=0? Does that mean f(x)=x^2 has only one root? Even though for every other value of x except zero there are two values for x that you can input to output a particular value for f(x). What about...
  38. A

    Finding the inverse of a polynomial in a field

    Homework Statement (A somewhat similar question to my last one). Let J be the ideal of the polynomial ring \mathbb{Q}[x] generated by x^2 + x + 3. Find the multiplicative inverse of (3x^3 + 3x^2 + 2x -1) + J in \mathbb{Q}[x]/JHomework Equations The Attempt at a Solution I think I need to apply...
  39. A

    Factorizing a polynomial over a ring

    Homework Statement Factorize x^2 + x + 8 in \mathbb{Z}_{10}[x] in two different waysHomework Equations The Attempt at a Solution I can see that x = 8 = -2 and x = 1 = -9 are roots of the polynomial, so one factorization is (x + 2)(x + 9). Is there a systematic way to find all the factorizations?
  40. C

    MHB Question to ponder about - - # of nonzero integer coefficients of a polynomial squared

    Suppose you look at polynomials, P(x), of degree n, with all nonzero integer coefficients and, in particular, a coefficient of 1 for the nth degree (leading) term. And look at those polynomials whose squares have the fewest number of nonzero integer coefficients possible. Examples...
  41. E

    Approximation Using Taylor POlynomial

    Find an approximate value of the number e-0.1 with an error less than 10-3 ı know that ex = Ʃ(from zero to ınfinity) xn / n!=1+x/1!+x 2/2!+... ı don't know how to use e-0.1 in this question.Do ı write -0.1 instead of x in ex series?
  42. S

    Find the equation of cubic polynomial

    Homework Statement If α, β and γ are roots of cubic polynomial and: αβγ = 6 α2+β2+γ2=20 α3+β3+γ3=121 Find the equation of cubic polynomial Homework Equations vieta The Attempt at a Solution The equation is in the form: x3 - (α+β+γ)x2 + (αβ + αγ + βγ) x - αβγ = 0 But I...
  43. Y

    Simplification of a rational polynomial function

    Now if I have a function y=(ab+ac)/a, it can be further factorised, y=(a(b+c))/a. Now if we cancel off the a, we will have only y=b+c that will also give the same y-values as the original form of the function y with respect to the same x-value. This statement implies that cancellation or...
  44. P

    Relation between coefficients and zeros of a quadratic polynomial

    Homework Statement For any quadratic polynomial ax2+bx+c having zeros β and α Prove that β + α = -b/a and αβ = c/a. Homework Equations The Attempt at a Solution I have found a method myself to prove α+ β = -b/a. However, I could not prove αβ = c/a. It goes like this. If α and β are the...
  45. C

    Rational Bezier Curve to Polynomial Bezier Curve Conversion?

    Is there a way to convert a rational bezier curve to a piecewise series of one or more polynomial bezier curves with minimal loss in accuracy, specially cubic ones? I've already tried searching the internet for pre-existing algorithms, but I haven't been able to find any usable results despite...
  46. S

    Evaluating a quartic polynomial.

    I want to find the root(for N) of this equation: \frac{(2N-1)^2}{N(1-N)}=Ce^t The hint says "consider taking a substitution u=N-1/2" ...which is the top bit of the fraction. But what does take a substitution here mean ? This is a part of a loooong modelling problem which involved an ugly...
  47. A

    Proving the Relationship between Operators on a Finite-Dimensional Space

    "Given operators σ,τ on a finite-dimensional space V, show that στ=i, and that σ=p(τ) for some polynomial p in F[x]." The first part was no problem. As for the second, I have a strong suspicion that p is the characteristic polynomial, mostly because I believe I heard of that fact before (that...
  48. N

    Is f(x,y) = a + bx + cy + dxy a Quadratic Polynomial?

    Hello, I just want to clear up a confusion. Is f(x,y) = a + bx + cy + dxy a quadratic polynomial? where x and y are variables and a,b,c,d are constants.
  49. D

    Should be simple polynomial integral

    This is the end of a triple integration problem. I can get down to what seems like it should be a simple polynomial integral of a single variable. Yet I just can't get the numbers to work out. \int_0^5 \frac{-1}{2}x^3 + \frac{15}{2} x^2 - \frac{75}{2} x + \frac{125}{2} dx The indefinite...
  50. S

    Does this method have a name? Function Approximation by Polynomial Sum

    I created a method for both approximating a function and extending a it's domain from a Natural to a Real Domain. Does this have a name already or any interesting application? Basically. Add polynomial of degree 0, 1, 2, 3, etc. Making at the same time the approximation function equal to f(0)...
Back
Top