algebra.theory.txt

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                                  ┃   ALGEBRA   ┃
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Algebraic sum:
  - algebraic expression with only addition|subtraction top-level operations
     - including none, i.e. every algebraic expression is an algebraic sum
  - term: each top-level operand
     - coefficient: parts of a term multiplying|dividing each other
        - numerical coefficient: when constant
     - number of terms:
        - 1|2|3: monomial|binomial|trinomial|...
        - 2+: multinomial

Polynomial:
  - notations: P(x,...) or P
  - when all exponents ∈ N
     - no negative exponent, i.e. no division where variable is denominator
     - no fractional exponent
  - degree:
     - of a term: sum of all exponents
        - unexponeniated variable: 1
        - constant: 0
     - of a polynomial: maximum term degree
        - named constant|linear|quadratic|... like exponeniated expressions
     - homogenous polynomial: when all terms have same degree
  - number of unique variables:
     - 1|2|3: univariate|bivariate|trivariate|...
     - 2+: multivariate
  - can always be expressed in canonical form: ∑ aₙ*x^n
     - often written in descending degrees
  - coefficient:
     - usually means constant coefficient a canonical form
     - integer|real|complex polynomial: when coefficients ∈ N|R|C

Algebraic fraction:
  - fraction x/y where x|y are algebraic expressions
  - simple|complex fraction: whether either x|y is fraction itself
  - in lowest terms: when cannot reduce fraction, i.e. lowest common factor between x|y is 1
  - integral expression:
     - when y is omitted
     - still an algebraic fraction
        - i.e. every algebraic expression is an algebraic fraction
     - mixed expression: algebraic sum of integral expressions and non-integral expressions
  - rational [algebraic] fraction|expression:
     - when x|y are polynomials
     - proper|improper: whether degree of x < y or not
  - irrational fraction:
     - when x|y contain fractional exponents