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Write the expanded form of (3a + 5b + 7c) 2 .
9a 2 + 25b 2 + 49c 2 + 30ab + 70bc + 52ac
9a 2 + 25b 2 + 49c 2 + 30ab + 70bc + 42ac
9a 2 + 25b 2 + 49c 2 + 30ab + 65bc + 42ac
9a 2 + 25b 2 + 49c 2 + 34ab + 70bc + 42ac
9a 2 + 25b 2 + 49c 2 + 30ab + 70bc + 42ac
The expression (3a + 5b + 7c)² is expanded using the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x=3a, y=5b, and z=7c results in (3a)² + (5b)² + (7c)² + 2(3a)(5b) + 2(5b)(7c) + 2(7c)(3a), which simplifies to 9a² + 25b² + 49c² + 30ab + 70bc + 42ac.
The expression (3a + 5b + 7c)² is expanded using the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x=3a, y=5b, and z=7c results in (3a)² + (5b)² + (7c)² + 2(3a)(5b) + 2(5b)(7c) + 2(7c)(3a), which simplifies to 9a² + 25b² + 49c² + 30ab + 70bc + 42ac.
This mathematical identity is a fundamental pillar of Algebra, which traces its historical development back to ancient Indian mathematicians like Brahmagupta and Bhaskara II, whose work in 'Bijaganita' laid the foundation for solving polynomial equations.
Algebraic identity for trinomial square: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx.
Coefficient calculation: 2 * 3a * 5b = 30ab, 2 * 5b * 7c = 70bc, 2 * 7c * 3a = 42ac.
Order of terms: Final expansion is a combination of squares of individual terms and twice the product of each unique pair.
Option A is incorrect because the final term 52ac is a calculation error (2 * 7 * 3 = 42, not 52).
Option C is incorrect due to the coefficient of bc being 65 instead of 70 (2 * 5 * 7 = 70).
Option D is incorrect because the first product 2 * 3a * 5b equals 30ab, whereas it lists 34ab.
Always double-check the coefficients resulting from the doubling (2x) step in polynomial expansions.
B is correct — The expanded form is 9a² + 25b² + 49c² + 30ab + 70bc + 42ac.
Algebraic identities are frequently tested in competitive exams; memorize (a+b)², (a-b)², (a+b+c)², and (a+b)³ to save time in simplification questions.
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