12.2 Single-Variable Like Terms

Single-Variable Like Terms applies the general structure of like terms to expressions containing only one distinct variable, focusing on how the exponent attached to that one variable determines which terms can be combined, and how coefficients of any sign, including implied and negative ones, combine correctly once like terms are identified.

Like Terms at Each Power Level

Like Linear Terms

Terms in which the single variable appears to the first power, with no exponent written, are like terms to one another, since they all share the identical variable part consisting of that variable raised to an exponent of one.

Like Squared-Variable Terms

Terms in which the single variable is raised to the second power are like terms to one another, since they all share the identical variable part consisting of that variable squared, distinct from terms containing the same variable at any other power.

Like Higher-Power Terms

Terms in which the single variable is raised to the same power greater than two are like terms to one another, following the identical pattern as linear and squared terms, matched purely by having the same exponent on the shared variable.

Different Powers as Unlike Terms

Terms containing the same single variable but raised to different exponents are not like terms, since their variable parts differ even though the variable symbol itself is identical; a linear term and a squared term of the same variable can never be combined into one term.

Variable Terms versus Constants

Variable Terms and Constants

A term containing the variable, at any power, is never a like term to a constant term, since a constant has no variable part at all while a variable term does; only constants are like terms to other constants, and only variable terms sharing the same power are like terms to each other.

Coefficients Involved in Single-Variable Like Terms

Implied Coefficients in Like Terms

A variable term written with no visible coefficient carries an implied coefficient of one, and this implied one is treated exactly as any other coefficient would be when identifying and later combining like terms of that variable.

Negative Coefficients in Like Terms

Like terms may carry negative coefficients, and the negative sign is simply part of that term's coefficient value; two like terms with opposite-sign coefficients remain like terms, differing only in the sign attached to their otherwise matching variable part.

Zero Coefficient Result

When like terms are eventually combined and their coefficients happen to sum to zero, the resulting coefficient of zero means the entire combined term vanishes from the expression, since a variable multiplied by a zero coefficient contributes nothing to the expression's value.