Question

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

Answer 1

Generally,

A cubic polynomial say, f(x) is of the form ax³ + bx² + cx + d.

And, can be shown w.r.t its relationship between roots as.

⇒ f(x) = k [x³ – (sum of roots)x² + (sum of products of roots taken two at a time)x – (product of roots)]

Where, k is any non-zero real number.

Here,
Sum of roots = 3
Sum of product of roots = -1
Product of roots = -3
so,

f(x) = k [x³ – (3)x² + (-1)x – (-3)]

∴ f(x) = k [x³ – 3x² – x + 3)]

where, k is any non-zero real number.