Polynomials Basics

An expression is a group of terms (numbers, variables, or both) showing a value, separated by + or -.
An equation is an expression set equal to a value.
Terms are separated by + or - operators.
Variables can be independent (cause) or dependent (effect).

Coefficient: A number multiplied by a variable (e.g., 4 in $4c$ ).
Constant: A fixed numerical value (e.g., 8 in $4c - 8 = d$ ).
Exponent: Indicates how many times a base is multiplied by itself (e.g., $a^2 = a \times a$ ).
Expression: A group of numbers and operators that indicates a value.
Equation: Indicates that two quantities have the same value through equal sign.
Operator: A symbol indicating a mathematical process (+, -, ×, ÷).
Term: A single number, a variable, or a combination of both.
Variable: A symbol (usually a letter) for an unknown value.

A single-variable polynomial is a sum of terms with different powers of the same variable.
- Examples: $y^2 - y + 6$ , $x + 2x + x^2$ , $18z^{10} + 2z + z - 14$
Monomials, binomials, and trinomials have one, two, and three terms, respectively.
The degree of a single-variable polynomial is the largest exponent of the variable.
Polynomials are also named based on their degree.
- Quadratic: Second-degree.
- Cubic: Third-degree.
- Quartic: Fourth-degree.
- Examples: $ax^2 + bx + c$ , $ax^3 + bx^2 + cx + d$ , $ax^4 + bx^3 + cx^2$
A polynomial with two variables is a sum of terms with different powers of the two variables.
- Examples: $y^2 - x + 6$ , $2xy + x + y^2$ , $2z^{14}y + 18z^{10} + y - 14$
The degree of a term in a multivariable polynomial is the sum of the exponents in that term.
The degree of a multivariable polynomial is the greatest sum of the exponents of the terms.

Terms are ordered by degree (descending) and like terms are combined.
Polynomials can have constants, variables, and positive or non-negative exponents.
Polynomials cannot have variables in the denominator of a fraction, fractional exponents, negative exponents, roots of variables, or an infinite number of terms.