Reading an Expression: Terms, Factors, Coefficients

In this lesson you will: Split an expression into terms, and split a term into factors; Read a coefficient with its sign attached; Say what each part of an expression means in a situation.

Terms are the pieces being added

Cut an expression at every + and every - that is not locked inside parentheses. What falls out are the terms.

So 7x^2-4x+9 has three terms: 7x^2, -4x and 9. The last one, a plain number, is the constant term — it is the only piece whose value does not move when x does.

Factors, and the number out front

A factor is something being multiplied. In the term 6(x+4) there are two factors, 6 and (x+4), and the whole thing is a single term — the plus sign is sealed inside the parentheses.

The coefficient is the numerical factor of a term. When no number is written, it is 1: the term x^2 has coefficient 1, and -y has coefficient -1.

Common mistake — Leaving the minus sign behind

Asked for the coefficient of x in 5-3x, a student answers 3.

The reason is honest enough. The eye reads the symbols left to right, sees a 3 sitting next to the x, and reports it. The minus sign looks like an instruction between two pieces rather than part of one of them.

But subtraction is addition of the opposite. The expression is 5+(-3x), so the second term is -3x and its coefficient is -3. That sign travels with the term through every later step — combining, substituting, factoring — and dropping it once poisons everything after.

How to catch it: Rewrite every subtraction as adding a negative before you read anything off. Then test: at x=2, 5-3x is -1, while a coefficient of +3 would predict 11.