Let p 1 x = x 3 - 2020 x 2 + b 1 x + c 1 and p 2 x = x 3 - 2021 x 2 + b 2 x + c 2 be polynomials having two common roots α and β . Suppose there exist polynomials q 1 x and q 2 x such that p 1 x q 1 x + p 2 x q 2 x = x 2 - 3 x + 2 . Then the correct identity is

Question

Let p 1 x = x 3 - 2020 x 2 + b 1 x + c 1 and p 2 x = x 3 - 2021 x 2 + b 2 x + c 2 be polynomials having two common roots α and β . Suppose there exist polynomials q 1 x and q 2 x such that p 1 x q 1 x + p 2 x q 2 x = x 2 - 3 x + 2 . Then the correct identity is, p 1 3 + p 2 1 + 4028 = 0, p 1 3 + p 2 1 + 4026 = 0, p 1 2 + p 2 1 + 4028 = 0, p 1 1 + p 2 2 + 4028 = 0

MARKS App · Accepted Answer

p 1 x q 1 x + p 2 x q 2 x = x 2 - 3 x + 2 p 1 x - p 2 x = x 2 + b 1 - b 2 x + c 1 - c 2 ⇒ q 1 x = 1 & q 2 x = - 1 p 1 x - p 2 x = x - 1 x - 2 t + 3 = 2020 ⇒ t = 2017 p 1 x = x - 1 x - 2 x - 2017 Similarly, p 2 x = x - 1 x - 2 x - 2018 (A) p 1 3 + p 2 1 + 4028 = 0 p 1 3 = - 4028 p 2 1 = 0 Hence it is true

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