Question

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

Answer 1

Hint: We know a general form of a cubic polynomial is $a{x^3} + b{x^2} + cx + d$. If $\alpha ,\beta $ and $\gamma $ are the zeroes of the cubic polynomial then $\alpha + \beta + \gamma = \dfrac{{ - b}}{a}$ , $\alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}$ and $\alpha \beta \gamma = \dfrac{{ - d}}{a}$. By putting the given values of the sum of zeroes, sum of the products of its zeroes taken two at a time and the product of its zeroes, we can get the value of $a,b,c$ and $d$. Then we can write the required cubic polynomial.

Complete step-by-step solution:
Given, the sum of zeroes, sum of the products of its zeroes taken two at a time and the product of its zeroes is $2,7$ and $14$.
Let any general cubic polynomial $a{x^3} + b{x^2} + cx + d$ and $\alpha $, $\beta $ and $\gamma $ are its zeroes.
Now, we can write
$\Rightarrow$$\alpha + \beta + \gamma = \dfrac{{ - b}}{a}$
$\Rightarrow$$\alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}$
$\Rightarrow$$\alpha \beta \gamma = \dfrac{{ - d}}{a}$
Now, the given values are
$\Rightarrow$$\alpha + \beta + \gamma = \dfrac{2}{1}$
$\Rightarrow$$\alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{7}{1}$
$\Rightarrow$$\alpha \beta \gamma = \dfrac{{14}}{1}$
Comparing the above equations. we get,
$\Rightarrow$$\dfrac{{ - b}}{a} = \dfrac{2}{1}$ , $\dfrac{c}{a} = \dfrac{7}{1}$ and $\dfrac{{ - d}}{a} = \dfrac{{14}}{1}$
Now, put $a = 1$ and then find the value of $b$, $c$ and $d$.
By solving the above equality. we get,
$b = - 2$ $c = 7$ and $d = - 14$
Now, put the values of $a$, $b$ , $c$ and $d$ in the general cubic equation to obtain a required cubic polynomial.

Thus, the required cubic polynomial is ${x^3} - 2{x^2} + 7x - 14$.

Note: Similar concept is applied to find a quadratic polynomial when the sum of its zeroes and product of its zeroes are given. For this suppose the general form of quadratic polynomial that is $a{x^2} + bx + c$ and we also know that $\alpha + \beta = \dfrac{{ - b}}{a}$ and $\alpha \beta = \dfrac{c}{a}$ where $\alpha $ and $\beta $ are its zeroes.