**ERIC Number:**EJ764385

**Record Type:**Journal

**Publication Date:**2007-Mar

**Pages:**6

**Abstractor:**Author

**ISBN:**N/A

**ISSN:**ISSN-0020-739X

**EISSN:**N/A

The Remainder Theorem and Factor Theorem for Polynomials over Noncommutative Coefficient Rings

Dobbs, D. E.

International Journal of Mathematical Education in Science and Technology, v38 n2 p268-273 Mar 2007

Let R be an associative ring which has a multiplicative identity element but need not be commutative. Let f(X) = a[subscript n]X[superscript n] + a[subscript n-1]X[superscript n-1] + ... + a[subscript 0] [is a member of] R[X] and [alpha] [is a member of] R. It is known that there exist uniquely determined q(X) = b[subscript n-1]X[superscript n-1] + ... + b[subscript 0] and p(X) = c[subscript n-1]X[superscript n-1] + ... + c[subscript 0] [is a member of] R[X] and r, s [is a member of] R such that f(X) = q(X) (X - [alpha]) + r = (X - [alpha])p(X) + s. Examples show that q(X) and p(X) need not coincide. As in the classical Remainder Theorem over a commutative ring, r = f([alpha]):+ a[subscript n][[alpha][superscript n]] + a[subscript n-1][[alpha][superscript n-1]] + ... + a[subscript 0], but s = [[alpha][superscript n]]a[subscript n] + [[alpha][superscript n-1]]a[subscript n-1] + ... + a[subscript 0]. As consequences, a pair of generalizations of the classical Factor Theorem is obtained, as well as a new characterization of commutative rings. This note could find classroom/homework use in a course on abstract algebra as enrichment material for the unit on rings and polynomials.

Descriptors: Reliability, Algebra, Mathematics Instruction, College Mathematics, Mathematical Applications, Mathematical Logic

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Higher Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A

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