Which of the following expressions is equivalent to $(a + b)^{3}$ for …

Question

Which of the following expressions is equivalent to $(a + b)^{3}$ for all real $a, b$?

Accepted Answer

B. $a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$

Standard algebraic identity (Identity 6 in the ETS Math Review):
$(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$.
- Trap A is the *most common* student error — assuming the exponent distributes over addition.
- Trap C drops the squared variables in the cross-terms.
- Trap D drops the coefficient of 3 on the cross-terms.
Check with $a = b = 1$: $(1 + 1)^{3} = 8$. RHS of B: $1 + 3 + 3 + 1 = 8$ ✓. RHS of A: $1 + 1 = 2$ ✗.

Answer

$a^{3} + b^{3}$

Answer

$a^{3} + 3ab + b^{3}$

Answer

$a^{3} + a^{2}b + ab^{2} + b^{3}$