How will ((a+b)^3-(a^3+b^3)) be written in factor form?
Answer and explanation
Correct answer: 3ab(a+b)
The governing concept is the cube identity followed by extraction of a common factor. Expand the binomial: (a+b)^3=a^3+3a^2b+3ab^2+b^3. Now subtract a^3+b^3. The pure cube terms cancel, leaving 3a^2b+3ab^2. Both remaining terms contain 3ab, so factor it out: 3a^2b+3ab^2=3ab(a+b). Thus option A is correct. Option B has an incorrect minus sign; expanding 3ab(a-b) gives 3a^2b-3ab^2, whereas both remaining terms here are positive. Option C omits the coefficient and variables produced by the expansion. Option D is also wrong because only the pure cubes cancel; the two middle terms remain.
Frequently asked questions
What is the correct answer to this question?
3ab(a+b)
Why is this the correct answer?
The governing concept is the cube identity followed by extraction of a common factor. Expand the binomial: (a+b)^3=a^3+3a^2b+3ab^2+b^3. Now subtract a^3+b^3. The pure cube terms cancel, leaving 3a^2b+3ab^2. Both remaining terms contain 3ab, so factor it out: 3a^2b+3ab^2=3ab(a+b). Thus option A is correct. Option B has an incorrect minus sign; expanding 3ab(a-b) gives 3a^2b-3ab^2, whereas both remaining terms here are positive. Option C omits the coefficient and variables produced by the expansion. Option D is also wrong because only the pure cubes cancel; the two middle terms remain.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Algebraic identities.
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