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Solve: (a + b)3 - (a - b)3.
2b3+6a2b
2a3+6ab2
6a2b+2b3
2b3+3a2b
2b3+6a2b
Remember the direct identity for the difference of cubes of binomials: (a+b)3−(a−b)3=2b(3a2+b2) to solve within seconds.
Algebraic expression to expand and simplify: (a+b)3−(a−b)3
(a+b)3−(a−b)3=2b3+6a2b
Remember the direct identity for the difference of cubes of binomials: (a+b)3−(a−b)3=2b(3a2+b2) to solve within seconds.
Forgetting to distribute the negative sign across all terms of (a−b)3, which results in sign errors during subtraction.
Expand the first binomial cube
Use the standard algebraic identity for the cube of a sum: (a+b)3=a3+3a2b+3ab2+b3.
(a+b)3=a3+3a2b+3ab2+b3
Expand the second binomial cube
Use the standard algebraic identity for the cube of a difference: (a−b)3=a3−3a2b+3ab2−b3.
(a−b)3=a3−3a2b+3ab2−b3
Subtract the expansions
Subtract the second expression from the first, noting that terms with a3 and 3ab2 cancel out while the others double.
(a3+3a2b+3ab2+b3)−(a3−3a2b+3ab2−b3)=6a2b+2b3
A is correct because expanding and simplifying the given expression yields 2b3+6a2b.
Recognizing standard algebraic identities quickly allows you to bypass lengthy manual multiplications in competitive exam quantitative sections.
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