Polynomials

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 $a_{0}, a_{1}, \dots, a_{d}$ . Furthermore, $a_{d} \neq 0$ .

Completing the Square

Completing the square is a technique for converting a quadratic polynomial of the form $a x^{2} + b x + c$ to the form $a (x - h)^{2} + k$ for some values of $h$ and $k$ .

Algebraic Numbers

The set of polynomials with coefficients in $Z$ , $Q$ , $R$ , and $C$ are denoted by $Z [x]$ , $Q [x]$ , $R [x]$ , and $C [x]$ respectively.

A number $x \in R$ is called an algebraic number if it satisfies $p (x) = 0$ , where $p$ is a non-zero polynomial in $Z [x]$ . Otherwise it is called a transcendental number.

Radicals

A radical refers to an $n$ th root of $x$ denoted as $\sqrt[n]{x}$ . The positive integer $n$ is called the index or degree. The number $x$ of which the root is taken is the radicand.

Rationalization is a process by which radicals are eliminated from an expression. An expression of form $\sqrt{a} \pm \sqrt{b}$ is rationalized by multiplying by its conjugate $\sqrt{a} \mp \sqrt{b}$ .

Bibliography

Fraleigh, John B. A First Course in Abstract Algebra. Seventh edition, Pearson new international edition. Harlow: Pearson, 2014.
Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).
Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.
Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Wikipedia. “Completing the square.” December 21, 2025. https://en.wikipedia.org/w/index.php?title=Completing_the_square.
Wikipedia. “nth root.” February 19, 2026. https://en.wikipedia.org/w/index.php?title=Nth_root.
Wikipedia. “Rationalisation (mathematics).” September 13, 2025. https://en.wikipedia.org/w/index.php?title=Rationalisation_(mathematics).