Math 2.0

definition

Root of a polynomial

Definition

Let RR be a ring and let f∈R[x]f \in R[x]. An element c∈Rc \in R is a root of ff if f(c)=0f(c) = 0, where f(c)f(c) is the value of ff at cc.

Remarks

The roots of x2−1x^2 - 1 in Z\mathbb{Z} are 11 and −1-1. The polynomial x2+1x^2 + 1 has no root in R\mathbb{R}. It has the roots ii and −i-i in C\mathbb{C}. By the factor theorem, over a commutative ring, cc is a root of ff if and only if x−cx - c divides ff.

Explanations

L1 human

A root is a solution of the equation f(x)=0f(x) = 0. Whether a polynomial has a root depends on the ring: x2−2x^2 - 2 has no root in Q\mathbb{Q}, but it has the root 2\sqrt{2} in R\mathbb{R}.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinition
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  1. Group concept
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Sources

  1. Web https://en.wikipedia.org/wiki/Zero_of_a_function, Polynomial roots

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