Question

Find a cubic polynomial with the sum, sum of the products of its zeroes taken two at a time, and the product of its zeroes as 2, 7, 14 respectively.

Answer 1

Hint: A cubic polynomial is the polynomial whose degree is 3 and it has 3 roots. We will use the sum, sum of the products and products given in the question to find the cubic polynomial.

Complete step-by-step answer:
Now, according to the theory of the polynomials, if the cubic expression is
f(x) = $a{x}^3+b{x}^2+cx+d$ and $\alpha ,\beta ,\gamma$ are the roots of this polynomial, then we have
sum of products = $\alpha +\beta +\gamma =-{\displaystyle \frac{b}{a}}$, where b is the coefficient of $x^2$ and a is the coefficient of $x^3$.
Also, we have a sum of products taken two at a time = $\alpha \beta +\beta \gamma +\gamma \alpha ={\displaystyle \frac{c}{a}}$, where c is the coefficient of x.
Product of zeroes = $\alpha \beta \gamma =-{\displaystyle \frac{d}{a}}$, where d is a constant term.
Now, we are given that sum = 2. Therefore,
${\displaystyle \frac{b}{a}}=-2$ or b = -2a.
Also, the sum of product of zeroes taken two at a time = 7. Therefore,
${\displaystyle \frac{c}{a}}=7$ or c = 7a.
Product of zeroes = 14. Therefore, ${\displaystyle \frac{d}{a}}=-14$ or d = -14a.
Now, we have f(x) = $a{x}^3+b{x}^2+cx+d$. So, putting value of b, c and d in the expression, we get
f(x) = $a{x}^3-2a{x}^2+7ax-14a$
Taking a common from the above expression, we get
f(x) = $a(x^3-2x^2+7x-14)$
So, the required polynomial is f(x) = $a(x^3-2x^2+7x-14)$.

Note: Whenever we come up with such types of questions, we will use the formula of sum, sum of product and product of zeroes to find any expression. Any polynomial expression can be found as we found the cubic polynomial. We have to follow a few steps to find the polynomial expression. First, we will let a polynomial expression, then we will use the formulas according to the values given in the question. Then we will find the values of variables using the formulas and put those values to get the desired polynomial expression.