Consider the following statements:<br><br>1. (ab+bc+ca) is a factor of a^{2}(b-c)^{3}+b^{2}(c-a)^{3}+c^{2}(a-b)^{3}.<br>2. (a+b+c) is a factor of a^{2}(b-c)^{3}+b^{2}(c-a)^{3}+c^{2}(a-b)^{3}.<br><br>Which of the statements given above is/are correct?
- A. Only 1 ✓
- B. Only 2
- C. Both 1 and 2
- D. Neither 1 nor 2
Correct Answer: A. Only 1
Explanation
The polynomial evaluates to 0 if a=b, b=c, or c=a, so (a-b)(b-c)(c-a) is a factor. As it is a degree-5 symmetric polynomial, factorization yields (a-b)(b-c)(c-a)(ab+bc+ca). Hence, (ab+bc+ca) is a factor, but (a+b+c) is not.
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