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wxrebecca
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What's the difference between polynomials (as elements of a ring of polynomials) and polynomial functions??
A nitpick; if x is a real-valued indeterminate, then those aren't functions from the reals. At least, 1/x is not. Depending on the specific language you are using, 1/x is either a partial function or a grammatical error.Jarle said:you can say that x*1/x = 1, but as functions from the reals they are not equal; the former is undefined at 0.
Hurkyl said:A nitpick; if x is a real-valued indeterminate, then those aren't functions from the reals. At least, 1/x is not. Depending on the specific language you are using, 1/x is either a partial function or a grammatical error.
mathwonk said:this is not correct over finite fields, i.e. they are only the same over infinite fields. I.e. over a finite field, the map from polynomials to polynomial functions has a huge kernel.
e.g. over Z/pZ, the non zero polynomial (x-1)(x-2)...(x-p) corresponds to the zero function on Z/pZ.
A polynomial is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, but no division. It can have one or more terms, and the degree of a polynomial is determined by the highest exponent of the variable.
A polynomial function is a mathematical function that is defined by a polynomial. It takes in an input value and produces an output value based on the coefficients and powers of the variable in the polynomial. It can be graphed as a curve on a coordinate plane.
The main difference is that a polynomial is an algebraic expression, while a polynomial function is a mathematical function that is defined by a polynomial. In other words, a polynomial is a mathematical object, while a polynomial function is a process or operation.
Yes, a polynomial can also be a polynomial function. This is because a polynomial function is defined by a polynomial, and a polynomial can be evaluated for specific input values to produce an output. However, not all polynomials are polynomial functions, as they may not necessarily represent a mathematical process.
Polynomials and polynomial functions have various applications in fields such as physics, engineering, economics, and statistics. They are used to model and analyze data, make predictions, and solve problems in these areas. For example, polynomials can be used to represent the trajectory of a projectile, while polynomial functions can be used to model population growth or economic trends.