2022 · Q9
Consider the following statements :
- 1(ab + bc + ca) is a factor of a²(b - c)³ + b²(c - a)³ + c²(a - b)³.2. (a + b + c) is a factor of a²(b - c)³ + b²(c - a)³ + c²(a - b)³.
Which of the statements given above is/are correct?
Elementary MathematicsAlgebra
- AOnly 1
- BOnly 2
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
The expression is antisymmetric and of degree 5, hence equals k(a−b)(b−c)(c−a)(ab+bc+ca). Substituting a=1,b=0,c=−1 gives 2 = k(1)(1)(−2)(−1), so k = 1. Thus (ab+bc+ca) is a factor (statement 1 is true) but (a+b+c) is not (statement 2 is false). The keyed option is A.