Question

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 3, –1 and –3 respectively.

Answer 1

A quadratic equation when sum and product of its zeros is given by:

f(x) = k{x³ – (sum of the zeros)x² + (sum of zeros taken two at a time)x² – (product of the zeros)}

where k is any non-zero real number. 

Here, Sum of zeroes = 3 sum of the product of its zeros taken two at a time = –1 product of its zeros = –3 

⇒ f(x) = k{x³ – (α + β + γ)x² + (αβ + βγ + γα)x – αβγ}

⇒ f(x) = k{x³ – 3x² – x + 3}

where k is any non-zero real number.