Monomials: coefficient, variable part and degree

The simplest building block

Every polynomial you'll meet in this course is built out of small pieces called monomials. A monomial is a single term: a number multiplying some variables, and nothing else.

These are monomials:

These are not:

A plain number such as \(12\) counts as a monomial too: it just has no variables.

Standard form

The product \(2x \cdot 5y \cdot x\) is a monomial, but a messy one. Before working with a monomial we write it in standard form: one number in front, and each variable appearing once, usually in alphabetical order.

A monomial in standard form has two parts:

• the coefficient is the numerical factor, including its sign;
• the variable part is everything else: the variables with their exponents.

Degree

The degree tells you how many variable factors a monomial contains.

A variable with no visible exponent has exponent 1. A variable that doesn't appear has degree 0 in that variable.

A nonzero constant like \(12\) has degree 0, since you can think of it as \(12x^0\) and \(x^0 = 1\). The number \(0\) itself is the zero monomial, and by convention it isn't given a degree.

Like terms

Comparing monomials is what makes adding them possible, which is the next lesson.

• \(6a^2b\) and \(-a^2b\) are like terms.
• \(6a^2b\) and \(6ab^2\) are not: same letters, but the exponents are swapped.
• \(3xy\) and \(-3yx\) are like terms, and they're also opposites: same variable part, opposite coefficients. Order doesn't matter, because multiplication is commutative.

Practice