Generalizing the Classical Remainder Theorem: A Reflection-Based Methodological Strategy
The framework of this paper is the presentation of a case study in which university students are required to extend a particular problem of division of polynomials in one variable over the field of real numbers (as generalizing action) clearly influenced by prior strategies (as reflection generaliza...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Foundations |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2673-9321/4/4/44 |
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| Summary: | The framework of this paper is the presentation of a case study in which university students are required to extend a particular problem of division of polynomials in one variable over the field of real numbers (as generalizing action) clearly influenced by prior strategies (as reflection generalization). Specifically, the objective of this paper is to present a methodology for generalizing the classical Remainder Theorem to the case in which the divisor is a product of binomials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo>)</mo></mrow><msub><mi>n</mi><mn>1</mn></msub></msup><msup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><msub><mi>a</mi><mn>2</mn></msub><mo>)</mo></mrow><msub><mi>n</mi><mn>2</mn></msub></msup><mo>⋯</mo><msup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub><mo>)</mo></mrow><msub><mi>n</mi><mi>k</mi></msub></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>a</mi><mi>k</mi></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>,</mo><msub><mi>n</mi><mn>2</mn></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mi>n</mi><mi>k</mi></msub><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>. A first approach to this issue is the Taylor expansion of the dividend <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> at a point <i>a</i>, which clearly shows the quotient and the remainder of the division of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>a</mi><mo>)</mo></mrow><mi>k</mi></msup></semantics></math></inline-formula>, where the degree of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>, say <i>n</i>, must be greater than or equal to <i>k</i>. The methodology used in this paper is the proof by induction which allows to obtain recurrence relations different from those obtained by other scholars dealing with the generalization of the classical Remainder Theorem. |
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| ISSN: | 2673-9321 |