2023 · Q42
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) :
- 1The coefficient of x² is 0.2. The coefficient of x is (a - b) (b - c) (c - a).
Which of the statements given above is/are correct?
Elementary MathematicsAlgebra
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
The x² coefficient is a(b−c) + b(c−a) + c(a−b) = 0, so statement 1 holds. The x coefficient is −[a(b²−c²) + b(c²−a²) + c(a²−b²)] = −(a−b)(b−c)(c−a) (verify with a=1,b=2,c=3: both sides equal −2), whereas statement 2 claims +(a−b)(b−c)(c−a) — the wrong sign. Only statement 1 is correct (A).