notebook/notes/algebra/polynomials.md

---
title: Polynomials
TARGET DECK: Obsidian::STEM
FILE TAGS: algebra::polynomial
tags:
  - algebra
  - polynomial
---

## Overview

Given nonnegative integer $d$, a **polynomial in $n$ of degree $d$** is a function $p(n)$ of the form $$p(n) = \sum_{i=0}^d a_i n^i$$
The coefficients of $p(n)$ are $a_0, a_1, \ldots, a_d$. Furthermore, $a_d \neq 0$.

%%ANKI
Basic
Using sigma notation, a polynomial $p(n)$ in $n$ of degree $d$ is a function of what form?
Back: $p(n) = \sum_{k=0}^d a_kn^k$ where $a_d \neq 0$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%

%%ANKI
Basic
What four algebraic operations are permitted in a polynomial?
Back: Addition, subtraction, multiplication, and exponentiation.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1713580808763-->
END%%

%%ANKI
Basic
What is $d$ in "a polynomial in $n$ of degree $d$"?
Back: $d$ is a nonnegative integer.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1713580808766-->
END%%

%%ANKI
Basic
What is $n$ in "a polynomial in $n$ of degree $d$"?
Back: The polynomial's indeterminate.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%

%%ANKI
Basic
Which coefficient is special in a polynomial and why?
Back: That attached to the monomial with highest degree because it cannot be zero.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%

%%ANKI
Basic
What are the coefficients of a polynomial?
Back: The constants of the monomials found in the polynomial.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%

%%ANKI
Basic
What name is given to a degree-0 polynomial?
Back: A constant.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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END%%

%%ANKI
Basic
What name is given to a degree-1 polynomial?
Back: A monomial.
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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END%%

%%ANKI
Basic
What name is given to a degree-2 polynomial?
Back: A binomial.
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).
<!--ID: 1708974221755-->
END%%

%%ANKI
Basic
What is a binomial?
Back: A polynomial containing two terms.
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).
<!--ID: 1708368078759-->
END%%

%%ANKI
Basic
What is polynomial fitting?
Back: The solving of a linear system to find the coefficients of a polynomial.
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).
<!--ID: 1713580109018-->
END%%

%%ANKI
Basic
Polynomial fitting is a strategy for discovering what part of a polynomial?
Back: The coefficients.
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).
<!--ID: 1713580808780-->
END%%

%%ANKI
Basic
What is the degree of a polynomial?
Back: The highest degree of the monomials with non-zero coefficients.
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).
<!--ID: 1713580109082-->
END%%

%%ANKI
Basic
How many data points are required to fit a polynomial?
Back: $k + 1$ where $k$ is the degree of the polynomial.
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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END%%

## Bibliography

* 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).
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).