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.">

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.

Given:

Sum, sum of the product of zeros taken two at a time, and product of the zeros are $3$, $-1$ and $-3$ respectively.

To do:

We have to find the cubic polynomial which satisfies the given conditions. 

Solution:


We know that, 

The standard form of a cubic polynomial is $ax^3+bx^2+cx+d$, where a, b, c and d are constants and $a≠0$.


It can also be written with respect to its relationship between the zeros as,

$f(x) = k[x^3 – (sum of roots)x^2 + (sum of products of roots taken two at a time)x – (product of roots)]$

Where, k is any non-zero real number.

Here,

$f(x) = k[x^3 – (3)x^2 + (-1)x – (-3)]$

$f(x) = k [x^3 – 3x^2 – x + 3]$

where, k is any non-zero real number is the required cubic polynomial. 

Updated on: 10-Oct-2022

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