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If a + b + c = 13 and ab + bc + ca = 4, then what is the value of ab(a + b) + bc(b + c) + ca(c + a) + 3abc?
38
9
17
52
52
The given expression ab(a + b) + bc(b + c) + ca(c + a) + 3abc factors into the product of (a + b + c)(ab + bc + ca). Substituting the given values 13 and 4, the result is 13 multiplied by 4, which equals 52.
The given expression ab(a + b) + bc(b + c) + ca(c + a) + 3abc factors into the product of (a + b + c)(ab + bc + ca). Substituting the given values 13 and 4, the result is 13 multiplied by 4, which equals 52.
This algebraic identity represents the expansion of cyclic expressions, a fundamental concept in Quantitative Aptitude for SSC CGL, CHSL, and Banking exams under the 'Algebraic Identities' module.
Algebraic identity used: ab(a + b) + bc(b + c) + ca(c + a) + 3abc = (a + b + c)(ab + bc + ca)
The expression is a cyclic polynomial symmetric in variables a, b, and c.
Understanding these identities is essential for rapid calculation in time-bound competitive exams.
The expansion (a+b)(b+c)(c+a) = (a+b+c)(ab+bc+ca) - abc is a related useful identity.
In competitive exams, recognizing patterns in cyclic expressions saves significant time compared to traditional substitution.
Option D is correct because 13 * 4 = 52, matching the result of the identity evaluation.
D is correct тАФ The evaluated result of the cyclic algebraic expression based on the provided values is 52.
Mastering algebraic identities often reduces the complexity of Number System problems, specifically those involving divisibility rules and remainder theorems.
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