What Are Polynomials?

A polynomial is an algebraic expression made up of one or more terms. Each term consists of a number (the coefficient), one or more variables, and possibly exponents.

For example:

$3 x + 5$ — a polynomial with 2 terms
$x^{2} - 2 x + 7$ — a polynomial with 3 terms
$4 a^{2} b - 3 ab + 6$ — a polynomial with 3 terms

Term

A single part of an algebraic expression, made up of a coefficient and variable(s). Terms are separated by + or − signs. For example, in $5 x^{2} - 3 x + 2$ , the three terms are $5 x^{2}$ , $- 3 x$ , and $2$ .

Coefficient

The numerical part of a term. In $- 3 x$ , the coefficient is $- 3$ . In $x$ , the coefficient is $1$ (it's just not written).

Constant Term

A term with no variable — just a number. In $3 x + 7$ , the constant term is $7$ .

Simplifying polynomials means writing them in the shortest, clearest form possible — just like simplifying a fraction such as $\frac{6}{8} = \frac{3}{4}$ .

Like Terms — The Foundation of Simplification

Like Terms

Terms that have exactly the same variable(s) raised to exactly the same power(s). Only like terms can be added or subtracted together.

Here are some examples of like and unlike terms:

Like Terms	Why?
$3 x$ and $- 2 x$	Both have the variable $x$ to the power 1
$5 a^{2}$ and $2 a^{2}$	Both have $a^{2}$
$4 x y$ and $- 7 x y$	Both have the same variables $x$ and $y$

Unlike Terms	Why NOT?
$3 x$ and $3 x^{2}$	Different powers of $x$
$5 a$ and $5 b$	Different variables
$4 x y$ and $4 x$	One has $y$ , the other doesn't

Analogy

Think of like terms like objects of the same type. You can add 3 apples and 2 apples to get 5 apples, but you can't add 3 apples and 2 oranges — they're different things! Similarly, $3 x + 2 x = 5 x$ , but $3 x + 2 y$ cannot be simplified further.

Warning

$3 x$ and $3 x^{2}$ are not like terms — the exponent must match, not just the variable letter!

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