Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.

Let, p(x) = ax³ + bx² + cx + d be a cubic polynomial whose zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.
Given that,
α + β + γ = 2
αβ + βγ + γα = -7
αβγ = -14
We know that,
α + β + γ = -(Cofficient of x²)/(Cofficient of x³)
αβ + βγ + γα = (Cofficient of x)/(Cofficient of x³)
αβγ = -(Constant term)/Cofficient of x³)
Therefore,
α + β + γ = -(Cofficient of x²)/(Cofficient of x³) = (-b)/a = 2/1
αβ + βγ + γα = (Cofficient of x)/(Cofficient of x³) = c/a = (-7)/1
αβγ = -(Constant term)/Cofficient of x³) = (-d)/a = (-14)/1
On comparing, a = 1, b = -2, c = -7 and d = 14
Hence, the required cubic polynomial is p(x) = x³ – 2x² – 7x + 14.

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