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(a−b)3+(b−c)3+(c−a)3=?, which of the following should come in place of? so that relation always holds true?
2(a−b)(b−c)(c−a)
(a+b+c)(a2+b2+c2−ab−bc−ca)
3(a−b)(b−c)(c−a)
(a−b)(b−c)(c−a)
3(a−b)(b−c)(c−a)
Test with values like a=2,b=1,c=0. Then (2−1)3+(1−0)3+(0−2)3=13+13+(−2)3=1+1−8=−6. The correct formula must evaluate to -6: 3(2−1)(1−0)(0−2)=3(1)(1)(−2)=−6.
An expression (a - b)3 + (b - c)3 + (c - a)3.
x3+y3+z3=3xyz if x+y+z=0
Test with values like a=2,b=1,c=0. Then (2−1)3+(1−0)3+(0−2)3=13+13+(−2)3=1+1−8=−6. The correct formula must evaluate to -6: 3(2−1)(1−0)(0−2)=3(1)(1)(−2)=−6.
Many students try to expand all cubes manually instead of using the conditional identity property, leading to massive calculation errors.
Check the sum of terms
Let x=(a−b), y=(b−c), and z=(c−a). Calculate the sum x+y+z.
(a−b)+(b−c)+(c−a)=a−a−b+b−c+c=0
Apply algebraic identity
Since the sum of the three terms is zero, we use the identity x3+y3+z3=3xyz.
(a−b)3+(b−c)3+(c−a)3=3(a−b)(b−c)(c−a)
C is correct because according to the identity for the sum of cubes where the sum of bases is zero, the result is equal to three times the product of the three terms.
This identity is a fundamental property used in competitive exams to simplify complex symmetric expressions quickly.
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