# Polynomials Definition and 24 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. ### I Questions about algebraic curves and homogeneous polynomial equations

It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~. In addition, if...
2. ### Challenge Math Challenge - August 2021

Summary: countability, topological vector spaces, continuity of linear maps, polynomials, finite fields, function theory, calculus 1. Let ##(X,\rho)## be a metric space, and suppose that there exists a sequence ##(f_i)_i## of real-valued continuous functions on ##X## with the property that a...
3. ### C/C++ Need help with C++ code

I'm not at all C++ literate, but I need to understand what a C++ code is doing for a math problem regarding the Stieltjes polynomials. Especially, I want to know what is happening to the "num" and "den" in the code; the code is in this link: Stieltjes
4. ### Minimizing as a function of variables

As promised, here is the original question, with the integral written in a more legible form.
5. ### I Is there a geometric interpretation of orthogonal functions?

Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...
6. J

### I Primes and Polynomials

Does there exist a polynomial P(x) with rational coefficients such that for every composite number x, P(x) takes an integer value and for every prime number x, P(x) does not take on an integer value? Can someone please guide me in the right direction? I've tried to consider the roots of the...
7. ### I Express power sums in terms of elementary symmetric function

The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials. I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...
8. ### I Can Taylor series be used to get the roots of a polynomial?

I'm using this method: First, write the polynomial in this form: $$a_nx^n+a_{n-1}x^{n-1}+...a_2x^2+a_1x=c$$ Let the LHS of this expression be the function ##f(x)##. I'm going to write the Taylor series of ##f^{-1}(x)## around ##x=0## and then put ##x=c## in it to get ##f^{-1}(c)## which will be...
9. ### Factoring a four term polynomial

Homework Statement I just want to know how get from ##4x^3+3x^2-6x-5=0 ## to ##(x+1)^2(4x-5)=0##. What's the trick when dealing with these nasty polynomials? I got the answer by taking another approach (solving a root equation) but I noted this is also a way to go, but I can't figure out the...

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...
11. ### 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...
12. ### 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...
13. ### 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) +...
14. ### 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)...
15. ### 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...
16. ### 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. [Broken] Find one basis of http://www.sosmath.com/CBB/latexrender/pictures/38d4e8e4669e784ae19bf38762e06045.png and...
17. ### 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...
18. ### Principle Ideals of a Polynomial Quotient Ring

Homework Statement Let A be the algebra \mathbb{Z}_5[x]/I where I is the principle ideal generated by x^2+4 and \mathbb{Z}_5[x] is the ring of polynomials modulo 5. Find all the ideals of A Let G be the group of invertible elements in A. Find the subgroups of the prime decomposition...
19. ### Linear Transformation and isomorphisms

Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...
20. ### Factor Theorem Question

Homework Statement If the polynomial P(x) = x^2+ax+1 is a factor of T(x)=2x^3-16x+b, find a, b Homework Equations The Attempt at a Solution Let (px+q) be a factor of P(x), p can possibly be 1 and so can q, according to factor theorem, Hence, factors (x+1) or (x-1) P(1) = 0, substituting I...
21. ### Highest degree of a given polynomial is

Homework Statement A polynomial p(x) is such that p(0)=5, p(1)=4, p(2)=9 and p(3)=20. the minimum degree it can have a) 1 b) 2 c) 3 d) 4 Homework Equations The Attempt at a Solution a) Not Possible can't connect these points using straight line b) Not even possible to connect these points...
22. ### Geometry question form ax^2+bx+c

I was tutoring a student and I came across the following question. I feel like I'm missing something obvious, but it seems like there are too many variables for an answer to be determined. The attached picture contains all of the question details.
23. ### Need Help With Factoring Polynomials

(x2)3(2x3)3/(4x)2 I got 1/2x^13, is this correct?
24. ### TextBooks for Some Topics in Mathematics

Hi, I need suggestions for picking up some standard textbooks for the following set of topics as given below: Ordinary and singular points of linear differential equations Series solutions of linear homogenous differential equations about ordinary and regular singular points...