Math 2.0

statement theorem

Repeated roots and the derivative

Statement

Let RR be a commutative ring, let f∈R[x]f \in R[x] with f≠0f \neq 0 and let c∈Rc \in R. Then cc is a root of ff with multiplicity at least 22 if and only if f(c)=0f(c) = 0 and f′(c)=0f'(c) = 0. Here f′f' is the formal derivative of ff.

Remarks

The proof: write f=(x−c)gf = (x - c) g. The product rule gives f′=g+(x−c)g′f' = g + (x - c) g', so f′(c)=g(c)f'(c) = g(c). And g(c)=0g(c) = 0 exactly when (x−c)2(x - c)^2 divides ff. For example, f=x3−3x+2=(x−1)2(x+2)f = x^3 - 3x + 2 = (x - 1)^2 (x + 2). Here f(1)=0f(1) = 0 and f′(1)=3−3=0f'(1) = 3 - 3 = 0.

Explanations

L1 human

At a repeated root, the graph of a real polynomial touches the axis and has a horizontal tangent. So the polynomial and its derivative are both zero there. The algebraic statement is true over every commutative ring.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionFormal derivative, definition, no evidence levelFormal derivativedefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionMultiplicity of a root, definition, no evidence levelMultiplicity of a rootdefinitionRepeated roots and the derivative, statement, no evidence levelRepeated roots and th…statement
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  1. Group concept
  2. Ring definition
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  4. Degree of a polynomial definition
  5. Formal derivative definition
  6. Polynomial evaluation definition
  7. Root of a polynomial definition
  8. Multiplicity of a root definition
  9. Repeated roots and the derivative statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Formal_derivative, Application to finding repeated factors

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