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Reading an Expression: Terms, Coefficients, Constants

Before you change an expression you have to see it clearly. Learn to break an expression into terms, spot each term's sign, and tell coefficients from constants from variables.

What an expression is

An algebraic expression is a recipe made of numbers, letters, and operations — for example 3x + 5 or 7 - 2a + a^2. There is no equals sign; an expression is a *value waiting to happen*, not a statement that two things are equal. A variable like x is a letter that stands for a number we may not know yet, and a plain number like 5 is a constant.

Cutting it into terms

A term is a chunk separated from its neighbours by + or - signs. The trick is to read each + or - as belonging to the term that *follows* it — that sign is part of the term. So in 7 - 2a + a^2 the terms are +7, -2a, and +a^2. The number multiplying the variable is the coefficient; a lone number with no variable is the constant term.

Expression:   7 - 2a + a^2

Term 1:  +7      constant term, coefficient — (none)
Term 2:  -2a     coefficient = -2,  variable = a
Term 3:  +a^2    coefficient = +1,  variable = a (squared)

Note: a^2 means 1 * a^2, so its coefficient is 1 even though
no number is written.  And -2a carries its minus sign.
Reading each term with its sign and coefficient attached.

A worked breakdown

  1. Take 5x^2 - x + 8 - 3xy. Scan left to right and mark every + and - as a cut.
  2. The terms are +5x^2, -x, +8, and -3xy. Carry each sign with its term.
  3. Coefficients: 5, -1, (none for the constant), -3. Constant term: 8. The -x hides a coefficient of -1.

Seeing terms this cleanly is the foundation for everything ahead: you can only combine what you can first name. In the next guide we use this skill to gather like terms.