The claim “8 and 8x are like terms.” is false. Like terms share the same variable part (same variables raised to the same powers); 8 is a constant term, while 8x is a variable term with \(x^1\).
Definition of like terms
Two terms are like terms when their variable parts match exactly. In one-variable algebra, terms of the form \(a x^n\) and \(b x^n\) are like terms because both contain \(x^n\), differing only in coefficients \(a\) and \(b\). Constants are like terms with other constants because they have no variable factor.
A constant can be written using \(x^0\) because \(x^0 = 1\) for \(x \neq 0\). This notation helps compare exponents when checking like terms.
Structure of 8 and 8x
The terms can be written in a parallel “coefficient × variable part” form:
\[ 8 = 8 \cdot 1 = 8 \cdot x^0 \qquad\text{and}\qquad 8x = 8 \cdot x = 8 \cdot x^1. \]
The coefficients match (both are 8), but the variable parts do not: \(x^0\) versus \(x^1\). Matching coefficients alone does not make terms like terms.
Consequences for simplification
Like terms combine by adding or subtracting coefficients while keeping the shared variable part unchanged. Since 8 and 8x do not share the same variable part, the expression \(8 + 8x\) remains a sum of unlike terms and cannot be simplified into a single term.
\[ 8 + 8x \neq 16x \qquad\text{and}\qquad 8 + 8x \neq 16. \]
Examples of like terms near this situation
Constants are like terms with constants, and \(x\)-terms are like terms with other \(x\)-terms:
| Expression | Like-term grouping | Combined form |
|---|---|---|
| \(8 + 5\) | constants | \(13\) |
| \(8x + 2x\) | \(x^1\)-terms | \(10x\) |
| \(8x + 8\) | unlike terms (\(x^1\) vs \(x^0\)) | \(8x + 8\) |
| \(8x^2 + 8x\) | unlike terms (\(x^2\) vs \(x^1\)) | \(8x^2 + 8x\) |
Common confusions
Matching numbers do not control like-term status; the variable structure controls it. The pair 8 and 8x shares a coefficient but not an exponent pattern. A helpful check is writing constants as \(x^0\) terms and confirming that every variable exponent matches exactly.