Let P x = x 2 + bx + c , where b and c are integers. If P ( x ) is a factor of both x 4 + 6 x 2 + 2 5 and 3 x 4 + 4 x 2 + 2 8 x + 5 , then the value of P ( 1 ) is

Step-by-step solution

If x = α is the common root of both the equations, then α 4 + 6 α 2 + 2 5 = 0 ⇒ α 4 = - 2 5 - 6 α 2 Using this value of α 4 in the second equation ⇒ 3 - 2 5 - 6 α 2 + 4 α 2 + 2 8 α + 5 = 0 ⇒ - 7 5 - 1 8 α 2 + 4 α 2 + 2 8 α + 5 = 0 ⇒ 2 α 2 - 4 α + 1 0 = 0 ⇒ α 2 - 2 α + 5 = 0 on comparing with x 2 + bx + c , we get b = - 2 , c = 5 So, P x = x 2 − 2x + 5 ⇒ P 1 = 1 - 2 + 5 = 4