Multiple choice

Let p(x) be a quadratic polynomial with constant term 1. Suppose $p(x)$ when divided by $x -1$ leaves remainder 2 and when divided by $x +1$ leaves remainder 4. Then the sum of the roots of $p(x) = 0$ is

  1. $-1$
  2. $1$
  3. $-\dfrac{1}{2}$
  4. $\dfrac{1}{2}$
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D Correct answer
Explanation

Let p(x) = ax^2 + bx + 1. Remainder theorem: p(1) = a + b + 1 = 2 => a + b = 1. p(-1) = a - b + 1 = 4 => a - b = 3. Adding: 2a = 4 => a = 2. Subtracting: 2b = -2 => b = -1. Sum of roots = -b/a = -(-1)/2 = 1/2.

AI explanation

Let the quadratic polynomial be p(x) = kx^2 + lx + 1, where the constant term is 1. Using the remainder theorem, p(1) = k + l + 1 = 2 and p(-1) = k - l + 1 = 4. Solving these two equations gives 2k = 4, so k = 2 and l = -1. The sum of the roots of p(x) = 0 is given by -l/k, which is -(-1)/2 = 1/2.

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