Polynomial Definition and 45 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 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...
2. B Why cubic?

Why is a third degree polynomial called a cubic polynomial? I just don’t see the connection between 3 and a cube.

34. 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...
35. A problem from polynomials

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 Equations The 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.
36. Confusion about eigenvalues of an operator

Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This...
37. Polynomial 3 degrees

Dear PF Forum, As we know in polynomial 2 degrees AX2 + BX + C = 0, there's a formula for solving it. What about 3 degrees for example: AX3 + BX2 + CX + D = 0, there's is really no formula for solving it? The only way to solve it is by hand? I have several methods in my head, at least...
38. Polynomial division homework

Homework Statement How many pairs of solutions make x^4 + px^2 + q = 0 divisable by x^2 + px + q = 0 Homework Equations x1 + x2 = -p x1*x2= q[/B] The Attempt at a Solution I tried making z = x^2 and replacing but got nowhere. I figure 0,1,-1 are 3 numbers that fit but I am not sure what's...

Just to double check, but if one wanted to, like in partial fraction decomposition, associate literal coefficients of polynomials with corresponding unknowns on the other side of the equation, the justification for this action is the definition of equality of polynomials? EDIT: I know this...
40. Python How can I input a polynomial equation of infinite terms in P

I have been given a task to create an interpolating/extrapolating programme. I have completed the programme for linear interpolation (2 points) but now must make it usable for 3 or more points, ie a polynomial of n points. I think I have the equation in general for a polynomial as it is an...
41. Can a cubic polynomial be solved without arccos?

I was reviewing the Cardano's method formula for a real cubic polynomial having 3 real roots. It seems that to do so, the arccos (or another arc*) of a term involving the p & q parameters of the reduced cubes must be done, and then followed by cos & sin of 1/3 of the result from that arccos -...
42. Can't remember how to solve equation with two variables

Umm from memory I used to use...that triangle: 1 1 1 1 2 1 1 3 3 1 Fibonachii was it? Pathetic I can't even remember the name. To factorise...or was it expand...polynomials...anyway, I don't think that's elevant here. My question is; I had an...
43. Question on polynomial orders

I am trying to use a numerical polynomial root finding method, but I am unsure of the order of an expression. For example, if I have something that looks like x2+5x √(x2+3)+x+1=0 what is the coefficient of the second order (and potentially even the first order) term? Is the entire 5x√... term...
44. N

Complex Polynomial of nth degree

Homework Statement Show that if P(z)=a_0+a_1z+\cdots+a_nz^n is a polynomial of degree n where n\geq1 then there exists some positive number R such that |P(z)|>\frac{|a_n||z|^n}{2} for each value of z such that |z|>R Homework Equations Not sure. The Attempt at a Solution I've tried dividing...
45. Degree of Polynomial

Can someone just confirm my answers to this easy polynomial question, State the degree and dominant term to f(x)=2x(x-3)^3(x-1)(4x-2) I am working on this online and there is nothing on working on equations like this in the lesson. I believe the degree to be either 2 or 6, as the functions end...