# 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.

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1. M

### Differentiating polynomial absolute value function

For this problem, The solution is, My solution is, Where I solved these equations to find where the function is ##f(x) > 0## and ##f(x) < 0##. Using ##x^2(1 - x) \geq 0## and ##x^2(1 - x) < 0## First equation: ##x^2 \geq 0 \implies x \geq 0## and ## 1 \geq x## Second equation: ##x^2(1...
2. ### How did they rewrite this polynomial in this way?

How can you rewrite polynomial in terms of (x-a) instead of x? One thing came to mind is rewrite each x as x-a+2 (So it is x-2+2 in our example) but this will take long time and a lot of algebra steps, how did they do it very fast in the attached picture? thanks
3. ### Determining the horizontal asymptote

Consider, I am self-studying; My interest is on the horizontal asymptote, now considering the degree of polynomial and leading coefficients, i have ##y=\dfrac{2}{1} =2## Therefore ##y=2## is the horizontal asymptote. The part that i do not seem to get is (i already checked this on desmos)...
4. ### A Quintic and Higher Degree Polynomial Equations

What's the root formula for fifth and higher degree polynomial equations, which have roots in radicals?
5. ### Roots of a polynomial mixed with a trigonometric function

When I look at the left hand side of the equation in above question then I can see that the highest degree of x would be 6 after the denominators are eliminated. I know that a polynomial of degree n will have n roots, but this one is not a pure polynomial since there is also a trigonometric...
6. ### I Solve the problem involving Rings

Its a bit clear; i can follow just to pick another polynomial say ##(x+1)^3## are we then going to have ##(2x-2)+ x+3##? or it has to be a polynomial with ##x^2+1## being evident? cheers...
7. ### A Differential equation and Appell polynomials

Hello! Let $n$ be a natural number, $P_n(x)$ be a polynomial with rational coefficients, and $\deg P_n(x) = n$. Let $P_0(x)$ be a constant polynomial that is not equal to zero. We define the sequence ${P_n(x)}_{n \geq 0}$ as an Appell sequence if the following relation holds: ...
8. ### Show that ##f(x)=2',1',2'## in the irreducible Polynomial

My interest is on the highlighted; my understanding is that, let ##f(x)=x^3+x^2+2^{'}## then ##f(1^{'})=1{'}+1{'}+2^{'}=4^{'} ## we know that in ##\mathbb{z_3} ## that ##\dfrac{4}{3}=1^{'}## ##f(2^{'})=8^{'}+4^{'}+2^{'}=14{'} ## we know that in ##\dfrac{14^{'}}{3}=1^{'}##... I hope...
9. ### 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~[1]. In addition, if...
10. ### I May I use set theory to define the number of solutions of polynomials?

Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely, $$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$, It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing...
11. ### A Computing the Minimal polynomial - Ring Theory

Am going through this notes...kindly let me know if there is a mistake on highlighted part. I think it ought to be; ##α^2=5+2\sqrt{6}##
12. ### Finding Integer Solutions to Polynomial Equations: Can it be Done Easily?

Hello, Am re-studying math & calculus aiming to start pure math studying later. However, I got this problem in Stewart calculus. Typically, this is a straightforward IVT application. x = x^3 + 1, call f(x)= x^3 - x + 1 & apply IVT. However I have two things to discuss. First thing is simple...
13. ### POTW Prove that the roots of a polynomial cannot be all real

Let ##a,\,b,\,c## and ##d## be any four real numbers but not all equal to zero. Prove that the roots of the polynomial ##f(x)=x^6+ax^3+bx^2+cx+d## cannot all be real.

15. ### A FEM basis polynomial order and the differential equation order

Is there a good rubric on how to choose the order of polynomial basis in an Finite element method, let's say generic FEM, and the order of the differential equation? For example, I have the following equation to be solved ## \frac{\partial }{\partial x} \left ( \epsilon \frac{\partial u_{x}...
16. ### Find the values of a and b in the given polynomial

My approach; Let ##(n+1)=k## ##(x-n+1)^3-(x-n)^3=3x^2+ax+b## ##(x-k)^3-(x-n)^3=3x^2+ax+b## ##(x^3-3x^2k+3xk^2-k^3)-(x^3-3x^2n+3xn^2-n^3)## ##=-3x^2k+3x^2n+3xk^2-3xn^2-k^3+n^3≡3x^2+ax+b## ##⇒1=n-k## ##⇒3(k^2-n^2)=a## ##⇒n^3-k^3=b## ##3(k+n)(k-n)=a## ##-3(k+n)=a## ##a=-3(n-1+n)## ##a=-3(2n-1)##...
17. ### How to check if a polynomial is irreducible over the rationals

I first checked for rational roots for this polynomial. The options are ##x=\pm 1/7##, both don't nullify the polynomial thus this polynomial doesn't have rational roots. Now, if it's reducible the only possible factors are: ##(ax^3+bx^2+cx+d)(Ax^3+Bx^2+Cx+D)=7x^6-35x^4+21x-1## or a product of...
18. ### B Why Is a Cubic Polynomial Called 'Third Degree'?

Why is a third degree polynomial called a cubic polynomial? I just don’t see the connection between 3 and a cube.
19. ### Can a polynomial have an irrational coefficient?

Is this a polynomial? y = x^2 + sqrt(5)x + 1 I was told NO, the coefficients had to be rational numbers. I this true? It seem to me this is an OK polynomial. I can graph it and use the quad formula to find the roots? so why or why not?

33. ### Why are there only two roots of this cubic polynomial?

Hi, I was trying to find roots of the following cubic polynomial and there are only two roots. I believe there should be three roots. Could you please guide me why there are only two roots? If you say that the "1" repeats itself as a root then I'd say the same could be said of "0.9". Thank...
34. ### Cramer's rule and first degree polynomial curve fitting

Hi, I did the first degree curve fitting in MATLAB. Please see below which also shows the output for each code line. But I wasn't able to get the same answer using Cramer's rule method presented below. I'm sure MATLAB answer is correct so where am I going wrong with the Cramer's rule method...
35. ### MHB Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$

Let $p,\,q$ and $r$ be the distinct roots of the polynomial $x^3-22x^2+80x-67$. It is given that there exist real numbers $A,\,B$ and $C$ such that $\dfrac{1}{s^3-22s^2+80s-67}=\dfrac{A}{s-p}+\dfrac{B}{s-q}+\dfrac{C}{s-r}$ for all $s\not \in \{p,\,q,\,r\}$. What is...
36. ### I Determine if polynomial system has finite number of solutions

Wish to determine when a system of polynomials has an infinite number of solutions, that is, is not zero-dimensional. The Wikipedia article : System of polynomial equations states: I interpret the quote to mean the system has an infinite number of solutions if the Grobner basis does not have...
37. ### MHB Solve Polynomial Division: -5a + 4b = x^2+1 Rem -A-2

The remainder of p(x)=x^3+ax^2+4bx-1 divided by x^2+1 is –5a + 4b. If the remainder of p(x) divided by x + 1 is –a – 2, the value of 8ab is ... A. -\frac34 B. -\frac12 C. 0 D. 1 E. 3 Dividing p(x) by x^2+1 by x^2+1 with –5a + 4b as the remainder using long division, I got (4bx – 1) – ((a – 1)x...
38. ### If a polynomial is identically zero, then all its coefficients are 0

Suppose ##a_0+a_1x+\ldots+a_nx^n=0## and restrict the domain of ##p## to the set of real numbers excluding the roots of ##p##. Note that: if ##a_0 == 0##: ##x=0## is a root of ##p## else: ##x=0## is not a root of ##p## Assume the latter. Subtract ##a_0## from both sides of the equation...

50. ### Find this sum involving a polynomial root

if x_{I}, I = {1,2,...,2019} is a root of P(x) = ##x^{2019} +2019x - 1## Find the value of ##\sum_{1}^{2019}\frac{1}{1-\frac{1}{X_{I}}}## I am really confused: This polynomial jut have one root, and this root is x such that 0 < x < 1, so that each terms in the polynomial is negative. But the...