The factor −3 must be distributed to every term inside the parentheses: −3 · 2x = −6x, and −3 · (−5) = +15. The result is −6x + 15. Option A is the most common error — distributing −3 to the first term but not properly applying the sign to the second. Option B results from treating −3 · (−5) incorrectly as −15. Remember: the sign is part of the factor and distributes like everything else; there is no protective barrier around the second term.
Question 2 Multiple Choice
A student needs to simplify 12x + 18. Which application of the distributive property produces the most fully factored form?
A2(6x + 9)
B3(4x + 6)
C6(2x + 3)
DThe distributive property only works left to right; it cannot be applied here
The distributive property runs in both directions. Applied right to left — ab + ac → a(b + c) — it is called factoring. The greatest common factor of 12x and 18 is 6, giving 6(2x + 3). Options A and B are partially factored but not completely: 6 and 9 share a common factor of 3, and 4 and 6 share a common factor of 2. Option D reflects a critical misconception — factoring is the same operation as distributing, just applied in reverse.
Question 3 True / False
The area model for the distributive property shows why every term inside the parentheses must be multiplied: each sub-rectangle requires the full width of the factor outside.
TTrue
FFalse
Answer: True
In the area model, a rectangle with width a and length (b + c) is divided into two sub-rectangles: one with area a·b and one with area a·c. The full width 'a' applies to every sub-rectangle — you cannot give part of the width to one and not to the other. This geometric fact is why a(b + c) = ab + ac holds: each term inside the parentheses receives the full factor. It also explains why distributing only to the first term — a(b + c) = ab + c — is geometrically nonsensical.
Question 4 True / False
The expression 3(x + 4) = 3x + 4 is a correct application of the distributive property.
TTrue
FFalse
Answer: False
This is the most common distributing error: multiplying 3 by the first term (x) but forgetting to multiply it by the second term (4). The correct expansion is 3(x + 4) = 3x + 12. Every term inside the parentheses must be multiplied by the factor outside. A useful check: count the terms inside (two: x and 4) and confirm the same number appear after distributing (two: 3x and 12).
Question 5 Short Answer
Explain why factoring — for example, rewriting 6x + 15 as 3(2x + 5) — is the same operation as the distributive property, not a separate rule.
Think about your answer, then reveal below.
Model answer: The distributive property states a(b + c) = ab + ac. Factoring applies this identity right to left: starting with ab + ac and recognizing it equals a(b + c). The mathematical relationship is identical; only the direction of application differs. Factoring is the distributive property run in reverse.
This conceptual unity is important because students who see factoring as a separate, mysterious operation struggle when they encounter it in advanced contexts like solving quadratic equations or simplifying rational expressions. Once you recognize that 6x + 15 = 3(2x + 5) is the same claim as 3(2x + 5) = 6x + 15 read from right to left, the operation becomes one unified tool rather than two separate ones.