An expression is arithmetic with a letter standing in for a number. The letter is defined before any symbolic rule uses it, then the expression is read into terms, signs, coefficients and factors without rearranging it.
Core ideas
Almost every error in “The Distributive Law”, “Combining Like Terms”, “Evaluating Expressions” and “Translating Words into Algebra” is an error of reading rather than of arithmetic: the wrong piece was treated as a unit.
It is not a new kind of object and it obeys no new rules. Everything in the arithmetic prerequisite and the reading convention of “Order of Operations” applies to x, because whatever number x is, those statements remain true.
That is the whole rule for finding them, and it is why brackets matter so much: a plus sign inside a bracket does not separate terms. 5(x + 2) is one term; 5x + 10 is two.
In 3x² − 5x + 7 the second term is −5x, not 5x. Subtraction is addition of a negative, so the expression is a sum of 3x², −5x and 7 — and reading it that way removes most sign errors before they happen.
The material
Variable, term, coefficient, factor and degree, each defined once and then used.
An expression is a recipe for a number: it says what to do, and given values for the letters it produces a value. It does not claim anything. 2x + 5 is an expression; 2x + 5 = 11 is an equation, which does make a claim and is Linear Equations and Inequalities's subject. The distinction matters immediately, because you can solve an equation and you cannot solve an expression.
A variable is a letter standing for a number whose value is not fixed. A constant is a number written as itself. An algebraic expression is any combination of variables and constants built with the operations of arithmetic.
The word variable is slightly misleading: within one calculation the letter stands for one number and does not change while you work. It varies across problems, not across lines.
The terms of an expression are the pieces it separates into at its top-level + and − signs. Top-level means not inside any bracket.
The sign immediately in front of a piece belongs to that piece. An expression is therefore always a sum of its terms, some of which are negative.
Writing the subtraction as an addition once, as above, is worth the line. “Properties of the Real Numbers” will license reordering these terms by commutativity of addition. For now the reading skill is narrower: if a term eventually moves, its attached sign must move with it.
A term that is a number times a product of variables splits into a coefficient — the numerical factor, sign included — and a variable part, the letters with their exponents.
In −5x the coefficient is −5 and the variable part is x. In (2/3)x² the coefficient is 2/3. A term with no letters at all, such as 7, is a constant term.
Two coefficients are written invisibly and are misread constantly. x means 1x, so its coefficient is 1. −x means (−1)x, so its coefficient is −1. Neither is zero and neither is absent; the 1 is simply not printed, because writing it would add nothing.
Terms are what an expression is added out of. Factors are what a term is multiplied out of. 4xy is a single term with factors 4, x and y. Confusing the two is what makes people cancel across a plus sign, and it causes real damage in Rational and Radical Expressions.
A bracket makes this visible. 5(x + 2) is one term, because there is no top-level + or − anywhere in it: it is a product of the two factors 5 and (x + 2). Multiply it out and you get 5x + 10, which is two terms. Both expressions have the same value for every x; “The Distributive Law” proves that with distribution. They do not have the same number of terms. So "how many terms" is a question about a particular way of writing, and the honest answer names the form: one as written, two expanded.
The degree of a term is the sum of the exponents on its variables. The degree of an expression is the largest degree of any of its terms, once like terms have been combined.
4xy has degree 1 + 1 = 2. A constant term such as 7 has degree 0, since 7 = 7x⁰.
The qualification about combining is a preview of “Combining Like Terms”. In 5x − 5x + 4 the highest degree written on the page is 1, but the two x terms sum to 0 and the expression is 4, of degree 0. Degree describes the expression, not the ink.
4xy − y² + 2 has three terms. Their coefficients are 4, −1 and 2; their degrees are 2, 2 and 0; the expression has degree 2. −x + (2/3)x² − 5 has three terms with coefficients −1, 2/3 and −5, and degree 2. The second is not written in descending order of degree, which is legal and sometimes inconvenient.
Interactive
Terms, coefficients, factors, degree
Method
Four passes. Do them in order; the bracket pass changes everything after it.
Ignore every sign inside a bracket. What is left between those signs are the terms, and their number is the number of terms as written.
A term preceded by − is a negative term. Once the signs are attached, the expression is a sum and the terms may be written in any order.
The coefficient carries the sign. A term with no printed number has coefficient 1 or −1; write it in if that helps, and then stop writing it in once it stops helping.
Add the exponents within each term, then take the largest — but only after combining like terms, because cancellation can lower it.
Nothing is being simplified here. This is reading, not working.
The coefficient −1 in the first line is the one people leave blank or call 0. It is neither. −x contributes −1 of an x to every sum it is in, and in “Combining Like Terms” that −1 has to be added to other coefficients like any other number.
The last line shows why degree is read after combining and not before. Nothing about 5x − 5x + 4 looks like a constant until the middle step is done.
Fade the table for −3a²b + 4b − 7: mark the top-level signs first, then supply the entries. It has three terms with coefficients −3, 4 and −7. Calling the first coefficient 3 leaves its sign behind; counting a² and b separately confuses factors with terms.
For −4x²y + y − 2(x + 1), name the terms as written, the sign and coefficient of each monomial term, and every factor of the bracketed term. Do not expand it: “The Distributive Law” owns that move. The reading is finished when another reader can point to the same pieces.
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