Let P(x) be a quadratic polynomial such that the sum of the roots of the equation P(x) = 0 is 5 . If P(1) + P(5) = 0 , then the product of the roots of the equation P(x) = 0 is equal to

Step-by-step solution

Let P(x) = ax^2 + bx + c . Given that the sum of the roots is 5 , we have: - b a = 5 b = -5a It is also given that P(1) + P(5) = 0 . Evaluating the polynomial at x = 1 and x = 5 gives: (a + b + c) + (25a + 5b + c) = 0 26a + 6b + 2c = 0 13a + 3b + c = 0 Substituting b = -5a into the above equation: 13a + 3(-5a) + c = 0 13a - 15a + c = 0 -2a + c = 0 c = 2a The product of the roots of P(x) = 0 is given by c a . Product of roots = 2a a = 2 . Answer: 2