Introduction to Polynomials: Terminology, Conventions, and Operations

A comprehensive vocabulary review deck covering polynomial classifications, standard formatting conventions, like terms rules, algebra tile modeling, and addition and subtraction rules.

Term (Polynomial)

A single number, variable, or product of numbers and variables with whole-number exponents (e.g., $3x^2$, $-2.5xy$, $7$, $abc^3$). Expressions with negative or fractional exponents, such as $3/x$ or $x^{-2}$, are disqualified.

Monomial

A polynomial consisting of exactly one term (e.g., $-5x^3$, $8xy$, $14$).

Binomial

A polynomial consisting of exactly two unlike terms (e.g., $2x - 5$, $4x^2 + 7$).

Trinomial

A polynomial consisting of exactly three unlike terms (e.g., $x^2 + 5x - 6$, $a^2 - 2ab + b^2$).

Descending Order of Degree

The standard algebraic convention requiring a polynomial expression to be arranged from the highest exponent power first, ending with the constant term.

Multi-Variable Term Convention

The algebraic standard of writing numerical coefficients first, followed by variables arranged in alphabetical order (e.g., $5x^2y$ rather than $y \cdot 5x^2$).

Coefficient

The numerical factor placed before and multiplying the variable(s) in an algebraic term (e.g., in $-5x^2 + 8x - 3$, the coefficient of $x^2$ is $-5$ and the coefficient of $x$ is $8$).

Constant Term

A standalone number in a polynomial expression that contains no variables and whose value does not change (e.g., $-3$ in $-5x^2 + 8x - 3$).

Degree of a Multi-Variable Polynomial

The highest exponent sum among all terms in the polynomial, found by summing the exponents of all variables within each individual term.

Like Terms

Terms that possess the identical variable base(s) raised to the exact same exponent(s). Their numerical coefficients may differ, and they are the only terms that can be combined via addition or subtraction.

Commutative Property of Multiplication (in Polynomials)

The property stating that the order in which factors are multiplied does not change the result, establishing that terms such as $abc$ and $bca$, or $x^3y^2$ and $y^2x^3$, are like terms.

Implicit Coefficient

The implied numerical multiplier of $1$ on any variable term that does not show a written coefficient (e.g., $f$ is identical to $1f$).

Sign Attribution Rule

The algebraic rule stating that the operational sign directly to the immediate left of each term belongs to that term permanently (e.g., in $4x - 5x^2$, the minus belongs to $5x^2$, making it $-5x^2$).

Algebra Tiles

Geometric area models where large squares represent $x^2$, elongated rectangles represent $x$, and small squares represent unit constants ($1$). Shaded tiles represent positive values and unshaded represent negative values.

Zero Pair

A pair consisting of one positive tile and one negative tile of equal size that combine to produce a net value of zero.

Removing Parentheses (Polynomial Addition)

The procedure where parentheses are dropped directly when adding polynomials because distributing a positive factor of $+1$ over terms does not change any signs: $+1(a + b) = +a + b$.

Crucial Subtraction Rule

The rule stating that to subtract a polynomial, change the operation from subtraction to addition and distribute $-1$ to every term inside the trailing parentheses, inverting every sign.

Phrasing Trap ("Subtract A from B")

The standard algebraic phrasing convention where the command "Subtract $A$ from $B$" dictates that the setup must be ordered as $B - A$.