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.

Given:

Sum, sum of the product of zeros taken two at a time, and product of the zeros are $2$, $-7$ and $-14$ 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,

$f(x) = k[x^3 – (2)x^2 + (-7)x – (-14)]$

$f(x) = k [x^3 – 2x^2 – 7x + 14]$

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

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Updated on: 10-Oct-2022

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