Find a cubic polynomial with the sum, sum of the product of its zeroes

1.	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.
Answer» Let the cubic polynomial be ax³ + bx³ + cx + d, and its zeroes be α, β and γ. Then, α + β + γ = 2 = \(\frac{-(-2)}{1}=\frac{-b}{a}\) αβ + βγ + γα = -7 = \(\frac{-7}{1}=\frac{c}{a}\) αβγ = – 14 = \(\frac{-14}{1}=\frac{c}{a}\) a = 1, then b = -2, c = -7 and d = 14. So, one cubic polynomial which satisfies the given conditions will be x³ – 2x² – 7x + 14.

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.

Answer»

Let the cubic polynomial be

ax³ + bx³ + cx + d, and its zeroes be α, β and γ.

Then,

α + β + γ = 2 = \(\frac{-(-2)}{1}=\frac{-b}{a}\)

αβ + βγ + γα = -7 = \(\frac{-7}{1}=\frac{c}{a}\)

αβγ = – 14 = \(\frac{-14}{1}=\frac{c}{a}\)

a = 1, then b = -2, c = -7 and d = 14.

So, one cubic polynomial which satisfies the given conditions will be x³ – 2x² – 7x + 14.