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. Mr Davis 97

    Linear independence of polynomials of different degree

    Homework Statement Let S be a set of nonzero polynomials. Prove that if no two have the same degree, then S is linearly independent. Homework EquationsThe Attempt at a Solution We will proceed by contraposition. Assume that S is a linearly dependent set. Thus there exists a linear dependence...
  2. PhotonSSBM

    B Finding Maxima/Minima of Polynomials without calculus?

    I'm tutoring a student who is in a typical precalculus/trig course where they're teaching her about graphing various arbitrary polynomials. Among the rules of multiplicity and intercepts they seem to be phrasing the questions such that they expect the students to also find the maxima and minima...
  3. Unteroffizier

    Studying Why Am I Struggling with Polynomials and Rational Expressions in Algebra II?

    Hello. I am currently studying at a specialized mechanical engineering high school. I'm in my first (or 10th) year, as I've stated before. I've done algebra I and algebra II, along with about one half of trigonometry (utilizing the trigonometric functions in practical problems, like splitting...
  4. MrsM

    Linear Algebra: characteristic polynomials and trace

    The question is : Is it true that two matrices with the same characteristic polynomials have the same trace? I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...
  5. karush

    MHB Approximating Functions with 3rd Order Taylor Polynomials

    $\tiny{242.13.1}$ $\textsf{a. Find the $3^{rd}$ Taylor polynomial for $\sec{x}$ at $a=0$}\\$ \begin{align} \displaystyle f^0(x)&=f(x)=\sec{x}\therefore f^0(0)=1\\ &=\frac{1}{0!} x^0=1 \\ f^1(x)&=(\sec{x})'=\tan{x}\sec{x} \therefore f^1(x)=0 \\ &=\frac{1}{0!} x^0+\frac{0}{1!} x^1=1+0=1 \\...
  6. karush

    MHB 11. 1.33-T nth order Taylor polynomials - - centered at a=100, n=0

    $\tiny {11. 1.33-T} $ $\textsf{Find the nth order Taylor polynomials of the given function centered at a=100, for $n=0, 1, 2.$}\\$ $$\displaystyle f(x)=\sqrt{x}$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n} \frac{f^{(k)}\left(a\right)}{k!}(x-a)^k$$ $\textsf{n=0}\\$...
  7. karush

    MHB Taylor Polynomials for $e^{-4x}$ at $x=0$

    $\tiny{206.11.1.16-T}$ $\textsf{Find the nth-order Taylor polynomials centered at 0, for $n=0, 1, 2.$}\\$ $$\displaystyle f(x)=e^{-4x}$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$ $\textsf{n=0}\\$ \begin{align} f^0(x)&\approx...
  8. karush

    MHB Find the nth-order Taylor polynomials centered at 0, for n=0, 1, 2.

    $\tiny{206.11.1.15-T}$ $\textsf{Find the nth-order Taylor polynomials centered at 0, for $n=0, 1, 2.$}$ \\ $$\displaystyle f(x)=cos(3x)$$ $\textsf{using}\\$ $$P_n\left(x\right) \approx\sum\limits_{k=0}^{n}\frac{f^{(k)}\left(a\right)}{k!}x^k$$ $\textsf{n=0}\\$...
  9. S

    Symmetries of graphs and roots of equations

    Is there a good way to relate the symmetries of the graphs of polynomials to the roots of equations? There's lots of material on the web about teaching students how to determine if the graph of a function has a symmetry of some sort, but, aside from the task of drawing the graph, I don't find...
  10. weezy

    I Proving Hermite polynomials satisfy Hermite's equation

    My book (by Mary L Boas) introduces first the Hermite differential equation for Hermite functions: $$y_n'' - x^2y_n=-(2n+1)y_n$$ and we find solutions like $$y_n=e^{x^2/2}D^n e^{-x^2}$$ where ##D^n=\frac{d^n}{dx^n}## Now she says that multiplying ##y_n## by ##(-1)^ne^{x^2/2}## gives us what is...
  11. lep11

    Prove two polynomials are equal in R^n

    Homework Statement The task is to prove that $$\lim_{x\rightarrow0}\frac{Q_1(x)-Q_2(x)}{\|x\|^k}=0 \implies Q_1=Q_2,$$ where ##Q_1,Q_2## are polynomials of degree ##k## in ##\mathbb{R}^n##. Homework Equations $$ \lim_{x\to 0} \frac{a x^\alpha}{\|x\|^n}=\left\{\begin{array}{c} 0 \textrm{ if }...
  12. dykuma

    Legendre Polynomials & the Generating function

    Homework Statement Homework Equations and in chapter 1 I believe that wanted me to note that The Attempt at a Solution For the first part of this question, as a general statement, I know that P[2 n + 1](0) = 0 will be true as 2n+1 is an odd number, meaning that L is odd, and so the Legendre...
  13. M

    MHB Proving Divisibility of Polynomials in Field Extensions

    Hey! :o Let $K/F$ be a field extension, $f,g\in F[X]$. I want to show that if $g\mid f$ in $K[X]$, then $g\mid f$ also in $F[X]$. Suppose that $g\mid f$ in $K[X]$. Then $f=g\cdot h$, where $h\in K[X]$. We have to show that $h\in F[X]$. Could you give me some hints how we could show that...
  14. N

    B Questions about equation roots

    At the beginning of the summer, I was studying a precalculus course, in which I was taught that whenever a polynomial equation has a root in the form a + sqrt(b)c or a + ib, then another root would be its conjugate, I took it for granted for that time and I thought it was intuitive. Later on...
  15. V

    Characteristics of Polynomials

    Homework Statement I have to sketch a graph of y=x(x-3)^2 Homework EquationsThe Attempt at a Solution I know that the zeros are 0 and 3. The part which confuses me is that end behaviours as well as turning points. I am unsure of which way the end behaviours should be pointing. Is the highest...
  16. T

    I Legendre polynomials and Rodrigues' formula

    Source: http://www.nbi.dk/~polesen/borel/node4.html#1 Differentiating this equation we get the second order differential eq. for fn, (1-x^2)f''_n-2(n-1)xf'_n+2nf_n=0 ....(22) But when I differentiate to 2nd order, I get this instead, (1-x^2)f''_n+2(n-1)xf'_n+2nf_n=0Applying General Leibniz...
  17. Celestion

    How do you divide polynomials in your head "on sight"?

    Homework Statement The question was to find the roots of x3 - 3x2 + 4x - 2 = 0 Homework Equations The first root is found by the factor theorem, substituting x=1 into the polynomial gives 0 therefore x=1 is one root and (x-1) is a factor. The Attempt at a Solution In the worked solution...
  18. D

    I Generating polynomials for a multistep method

    Hi, I'm struggling to understand how the generating polynomials work and are implemented in the difference equation for a general ODE y' = f(t,y) Difference Equation Generating polynomials "Coefficients are normalized either by a_k = 1 or sigma(1) = 1
  19. J

    I Lemma regarding polynomials with one for all coefficients

    I managed to get through all of the problems in chapter three of Problem Solving through Problems, and I am now on to chapter 4 in the section on polynomials. A few problems I have encountered so far involve polynomials of the form: P(x) = 1 + x + x2 + x3 +...+xn I noticed that when n, the...
  20. S

    I If pair of polynomials have Greatest Common Factor as 1 ....

    NOTE: presume real coefficients If a pair of polynomials have the Greatest Common Factor (GCF) as 1, it would seem that any root of one of the pair cannot possibly be a root of the other, and vice-versa, since as per the Fundamental Theorem of Algebra, any polynomial can be decomposed into a...
  21. S

    I Matrix of columns of polynomials coefficients invertibility

    I am reviewing the method of partial fraction decomposition, and I get to the point that I have a matrix equation that relates the coefficients of the the original numerator to the coefficients of the numerators of the partial fractions, with the each column corresponding to a certain polynomial...
  22. kyphysics

    I Question About Long Division of Polynomials

    Dividend: 4x^3 - 6x - 11 Divisor: 2x - 4 In this problem above, the dividend lacks a variable to the second power, so we have to add a 0x^2 to make it: 4x^3 + 0x^2 - 6x - 11 Question: Why do we add 0x^n? (n = missing powers) In regular long division, we do no such thing. Why do we have...
  23. avikarto

    I Associated Legendre polynomials: complex vs real argument

    I am having trouble understanding the relationship between complex- and real-argument associated Legendre polynomials. According to Abramowitz & Stegun, EQ 8.6.6, $$P^\mu_\nu(z)=(z^2-1)^{\mu/2}\cdot\frac{d^\mu P_\nu(z)}{dz^\mu}$$ $$P^\mu_\nu(x)=(-1)^\mu(1-x^2)^{\mu/2}\cdot\frac{d^\mu...
  24. S

    A tricky remainder theorem problem

    Homework Statement A polynomial P(x) is divided by (x-1), and gives a remainder of 1. When P(x) is divided by (x+1), it gives a remainder of 3. Find the remainder when P(x) is divided by (x^2 - 1) Homework Equations Remainder theorem The Attempt at a Solution I know that P(x) = (x-1)A(x) +...
  25. Math Amateur

    MHB Polynomials in n indeterminates and UFDs

    In the introduction to Chapter 1 of his book "Introduction to Plane Algebraic Curves", Ernst Kunz states that the polynomial ring K[ X_1, X_2, \ ... \ ... \ X_n] over a field K is a unique factorization domain ... ... but he does not prove this fact ... Can someone demonstrate a proof of this...
  26. Math Amateur

    I Polynomials in n indeterminates and UFDs

    In the introduction to Chapter 1 of his book "Introduction to Plane Algebraic Curves", Ernst Kunz states that the polynomial ring ##K[ X_1, X_2, \ ... \ ... \ , X_n]## over a field ##K## is a unique factorization domain ... ... but he does not prove this fact ... Can someone demonstrate a proof...
  27. 5

    Help with finding Zeros of a polynomial with 1+i as a zero

    Homework Statement p(x) = x^3 − x^2 + ax + b is a real polynomial with 1 + i as a zero, find a and b and find all of the real zeros of p(x).The Attempt at a Solution [/B] 1-i is also a zero as it is the conjugate of 1+i so (x-(1+i))(x-(1-i))=x^2-2x+2 let X^3-x^2+ax+b=x^2-2x+2(ax+d)...
  28. T

    Proving theorem for polynomials

    Homework Statement Prove the following statement: Let f be a polynomial, which can be written in the form fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0 and also in the form fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0 Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n Homework Equations 3. The...
  29. V

    What is the common root for two polynomial equations with a shared coefficient?

    Homework Statement [/B] Th value of 'a' for which the equation x3+ax+1=0 and x4+ax+1=0 have a common root is?Homework EquationsThe Attempt at a Solution i initially thought of subtracting both the equations and then finding x and substituting back in the equation but it did not work.
  30. paulmdrdo1

    MHB Factoring Polynomials: A Faster Way!

    Hello! I just want to know the faster way to factor the expression. I already factored it out using trial and I am hoping you could give me some tricks to go about it faster than the usual method. Thanks $192x^3-164x^2-270x$
  31. G

    Matrix of linear transformation

    Homework Statement Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0...
  32. L

    MHB A whole function approximating polynomials

    Given a series of polynomials p_{n} and a series of open, non-intersecting sets V_{n} \subset \mathbb{C} show that there exists a function g\in \mathcal{O}(\mathbb{C}) such that lim_{n \rightarrow \infty} sup_{z \in V_{n}} |g(z)-p_{n}(z)|=0. Normally the approximation goes the other way around...
  33. M

    MHB Prove: Q(ζp)=Q(ζip) | Cyclotomic Polynomials

    Let ζp be e2πi/p. For an integer i, such that p does not divide i, prove Q(ζp) = Q(ζip ). I think this has something to do with both exponents of ζp (1 and i) being coprime to p, but I am not sure at all how to show the equality. If anyone could please help with an explanation, that would be...
  34. G

    I Solving complex formulas and higher order polynomials

    How would you go about solving (4(x^3)+38(x^2)+44x-20)/(20+12x+x^2), without the use of a computer, further, what about functions which have more x components, with higher powers. Also what process do computers use to solve these.
  35. M

    MHB Field Extensions for polynomials

    Describe the multiplication in the ring Q[x]/(x2+x+1). Is this a field? What is the multiplicative inverse of [x]? In describing the multiplication, would I just be describing something in regards to the multiplicative properties of a ring? i.e: a(bc)=(ab)c a(b+c)=ab+ac a*1=1*a=a ab=ba Is it...
  36. G

    MHB Subspace spanned by subsets of polynomials

    In the linear space of all real polynomials $p(t)$, describe the subspace spanned by each of the following subsets of polynomials and determine the dimension of this subspace. (a) \left\{1,t^2,t^4\right\}, (b) \left\{t,t^3,t^4\right\}, (c) \left\{t,t^2\right\}, (d) $\left\{1+t, (1+t)^2\right\}$...
  37. M

    MHB Symmetric Polynomials s1,s2,s3

    Express r12+r22+...+rn2 as a polynomial in the elementary symmetric polynomials s1, s2, . . . ,sn. I'm sure the equation we are dealing with is (r1+r2+...+rn)2 which is very large to factor out but should yield r12+r22+...+rn2+(other terms) I believe s1=r1+r2+...+rn s2=Σri1ri2 for...
  38. M

    MHB Symmetric Polynomials Involving Discriminant Poly

    Question: Let τ = (i, j) ∈ Sn with 1 ≤ i < j ≤ n. Prove: δ(rτ(1) , . . . ,rτ(n) ) = −δ(r1, . . . ,rn) Note: Discriminant Polynomial δ(r1,r2,...,rn) = ∏ (ri - rj) for i<j I am pretty confused on where to begin. Based on the note, does −δ(r1, . . . ,rn) then =...
  39. RJLiberator

    Factoring Polynomials [Abstract Algebra]

    Homework Statement 1. Let g(x) = x^4+46. a) Factor g(x) completely in ℚ[x]. b) Factor g(x) completely in ℝ[x]. c) Factor g(x) completely in ℂ[x]. 2. Completely factor the given polynomial in ℤ_5. [4]_5 x^3 + [2]_5 x^2 + x + [3]_5 Homework Equations ℚ = {m/n / m and n belong to Z, m is not...
  40. HaLAA

    Find all irreducible polynomials over F of degree at most 2

    Homework Statement Let F = {0,1,α,α+1}. Find all irreducible polynomials over F of degree at most 2. Homework EquationsThe Attempt at a Solution To determine an irreducible polynomial over F, I think it is sufficient to check the polynomial whether has a root(s) in F, So far, I got...
  41. RJLiberator

    Abstract algebra Polynomials and Prime

    Homework Statement Let g(x) ∈ ℤ[x] have degree at least 2, and let p be a prime number such that: (i) the leading coefficient of g(x) is not divisible by p. (ii) every other coefficient of g(x) is divisible by p. (iii) the constant term of g(x) is not divisible by p^2. a) Show that if a ∈ ℤ...
  42. nmsurobert

    3rd and 4th hermite polynomials

    Homework Statement Calculate the third and fourth hermite polynomials Homework Equations (1/√n!)(√(mω/2ħ))n(x - ħ/mω d/dx)n(mω/πħ)1/4 e-mωx2/2ħ ak+2/ak = 2(k-n)/((k+2)(k+1)) The Attempt at a Solution i kind of understand how how to find the polynomials using the first equation up to n=1. I'm...
  43. RJLiberator

    Multiplicative Inverse of Polynomials with Integer Coeff.

    Homework Statement If ℤ[x] denotes the commutative ring consisting of all polynomials with integer coefficients, list all the elements in ℤ[x] that have a multiplicative inverse in ℤ[x]. Homework Equations Multiplicative inverse if rs = 1 where rs ∈ R (rs are elements of the ring)...
  44. J

    MHB Divide Polynomials: (-2z^3-z+z^2+1) by (z+1)

    i am having trouble dividing (-2z^3-z+z^2+1)/ (z+1). Can someone please help?
  45. G

    Linear algebra: Finding a basis for a space of polynomials

    Homework Statement Let and are two basis of subspaces and http://www.sosmath.com/CBB/latexrender/pictures/69691c7bdcc3ce6d5d8a1361f22d04ac.png. Find one basis of http://www.sosmath.com/CBB/latexrender/pictures/38d4e8e4669e784ae19bf38762e06045.png and...
  46. H

    Proof of the Rodrigues formula for the Legendre polynomials

    How is the below expression for ##a_{n-2k}## motivated? I verified that the expression for ##a_{n-2k}## satisfies the recurrence relation by using ##j=n-2k## and ##j+2=n-2(k-1)## (and hence a similar expression for ##a_{n-2(k-1)}##), but I don't understand how it is being motivated. Source...
  47. H

    Verify the Rodrigues formula of the Legendre polynomials

    How does (6.79) satisfy (6.70)? After substitution, I get $$(1-w^2)\frac{d^{l+2}}{dw^{l+2}}(w^2-1)^l-2w\frac{d^{l+1}}{dw^{l+1}}(w^2-1)^l+l(l+1)\frac{d^{l}}{dw^{l}}(w^2-1)^l$$ Using product rule in reverse on the first two terms...
  48. D

    Find the set that contains the real solution to an equation

    Homework Statement ##\frac{x+\sqrt3}{\sqrt{x}+\sqrt{x+\sqrt3}} + \frac{x-\sqrt3}{\sqrt{x}-\sqrt{x-\sqrt3}} = \sqrt{x}## All real solutions to this equation are found in the set: ##a) [\sqrt3, 2\sqrt3), b) (2\sqrt3, 3\sqrt3), c) (3\sqrt3, 6), d) [6, 8)## Homework Equations 3. The Attempt at a...
  49. Math Amateur

    MHB Polynomials Acting on Spaces - B&K Ex. 1.2.2 (iv): An Intro by Peter

    I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help in order to fully understand Example 1.2.2 (iv) [page 16] ... indeed, I am somewhat overwhelmed by this construction ... ... Example 1.2.2 (iv) reads as follows:My...
  50. K

    MHB Prove that there are infinitely many irreducible polynomials in Z5.

    Here is my first (incorrect) attempt at a proof: Assume that there are finitely many (say n) irreducible polynomials in \(\mathbb{Z}_5\). Let \(a(x)=p_1(x)p_2(x)...p_i(x)...p_n(x) +1\) where \(p_i(x)\) is the i-th irreducible polynomial in \(\mathbb{Z}_5\). a(x) is irreducible and not equal...
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