130 lines
4.9 KiB
Markdown
130 lines
4.9 KiB
Markdown
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---
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title: Polynomials
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::polynomial
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tags:
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- algebra
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- polynomial
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---
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## Overview
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Given nonnegative integer $d$, a **polynomial in $n$ of degree $d$** is a function $p(n)$ of the form $$p(n) = \sum_{i=0}^d a_i n^i$$
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The coefficients of $p(n)$ are $a_0, a_1, \ldots, a_d$. Furthermore, $a_d \neq 0$.
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%%ANKI
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Basic
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Using sigma notation, a polynomial in $n$ of degree $d$ is a function of what form?
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Back: $p(n) = \sum_{i=0}^d a_in^i$ where $a_d \neq 0$.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808758-->
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END%%
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%%ANKI
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Basic
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What four algebraic operations are permitted in a polynomial?
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Back: Addition, subtraction, multiplication, and exponentiation.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808763-->
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END%%
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%%ANKI
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Basic
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What is $d$ in "a polynomial in $n$ of degree $d$"?
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Back: $d$ is a nonnegative integer.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808766-->
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END%%
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%%ANKI
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Basic
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What is $n$ in "a polynomial in $n$ of degree $d$"?
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Back: The polynomial's indeterminate.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808769-->
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END%%
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%%ANKI
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Basic
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Which coefficient is special in a polynomial and why?
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Back: That attached to the monomial with highest degree because it cannot be zero.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808772-->
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END%%
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%%ANKI
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Basic
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What are the coefficients of a polynomial?
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Back: The constants of the monomials found in the polynomial.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1713580808776-->
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END%%
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%%ANKI
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Basic
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What name is given to a degree-0 polynomial?
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Back: A constant.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1708974221749-->
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END%%
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%%ANKI
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Basic
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What name is given to a degree-1 polynomial?
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Back: A monomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708974221752-->
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END%%
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%%ANKI
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Basic
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What name is given to a degree-2 polynomial?
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Back: A binomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708974221755-->
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END%%
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%%ANKI
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Basic
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What is a binomial?
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Back: A polynomial containing two terms.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708368078759-->
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END%%
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%%ANKI
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Basic
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What is polynomial fitting?
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Back: The solving of a linear system to find the coefficients of a polynomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109018-->
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END%%
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%%ANKI
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Basic
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Polynomial fitting is a strategy for discovering what part of a polynomial?
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Back: The coefficients.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580808780-->
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END%%
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%%ANKI
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Basic
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What is the degree of a polynomial?
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Back: The highest degree of the monomials with non-zero coefficients.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109082-->
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END%%
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%%ANKI
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Basic
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How many data points are required to fit a polynomial?
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Back: $k + 1$ where $k$ is the degree of the polynomial.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1713580109089-->
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END%%
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## Bibliography
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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