Consider the following statements in respect of the polynomial a(b-c)(x-b)(x-c)+b(c-a)(x-c)(x-a)+c(a-b)(x-a)(x-b) :<br>1. The coefficient of x^{2} is 0.<br>2. The coefficient of x is (a-b)(b-c)(c-a).<br>Which of the statements given above is/are correct?
- A. 1 only
- B. 2 only
- C. Both 1 and 2 ✓
- D. Neither 1 nor 2
Correct Answer: C. Both 1 and 2
Explanation
The coefficient of x^2 is a(b-c) + b(c-a) + c(a-b) = 0, so statement 1 is correct. The coefficient of x is -a(b-c)(b+c) - b(c-a)(a+c) - c(a-b)(a+b) = -[ab^2-ac^2+bc^2-ba^2+ca^2-cb^2], which factorizes exactly to (a-b)(b-c)(c-a). So statement 2 is also correct.
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