Which statement accurately defines when two algebraic expressions are equivalent?

• A. They have the same degree.
• B. They have the same value for all variable assignments.
• C. They have the same number of terms.
• D. They have the same variables and coefficients.

Answer: (B)

The main idea here is what it means for two algebraic expressions to be equivalent: they give the same value no matter what numbers you substitute for the variables. If you can replace one expression with the other in any equation and keep the equation true for every possible input, the two expressions are equivalent.

Two expressions can share the same degree, or have the same number of terms, or involve the same variables with similar-looking coefficients, but that doesn’t guarantee they produce the same values for all inputs. For example, expressions with the same degree can yield different results when you plug in numbers (like x+2 and 2x+2; at x=1 they give 3 and 4). Two expressions with the same number of terms can also give different values for some x (such as x+2 versus 3x+4). And just having the same variables and coefficients doesn’t by itself ensure the expressions are the same for all inputs unless they simplify to the same expression. The precise criterion for equivalence is that their values match for every possible assignment of the variables.