Question

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

Answer 1

Step-1 : Given data :

Let $\alpha ,\beta ,\gamma$ be the roots(or zeros) of the required cubic polynomial.

Then, according to the given data,

Sum of the zeros $=\alpha +\beta +\gamma =\frac{2}{1}$

Sum of the product of its zeros taken two at a time $=\alpha \beta +\beta \gamma +\gamma \alpha =-\frac{7}{1}$

Product of zeros $=\alpha \beta \gamma =-\frac{14}{1}$

Step-2 : Finding the coefficients of the cubic polynomial:

We know that the general form of a cubic polynomial is $a{x}^3+b{x}^2+cx+d$ and the relation between the sum and product of its zeroes and coefficients of the polynomial is

$\mathit{\alpha }\mathbf{+}\mathit{\beta }\mathbf{+}\mathit{\gamma }\mathbf{=}\mathbf{-}\frac{\mathbf{b}}{\mathbf{a}}$

$\mathit{\alpha }\mathit{\beta }\mathbf{+}\mathit{\beta }\mathit{\gamma }\mathbf{+}\mathit{\gamma }\mathit{\alpha }\mathbf{=}\frac{\mathbf{c}}{\mathbf{a}}$

$\mathit{\alpha }\mathit{\beta }\mathit{\gamma }\mathbf{=}\mathbf{-}\frac{\mathbf{d}}{\mathbf{a}}$

Thus, on comparing the coefficients, we get, $a=1$, $b=–2$, $c=–7$ and $d=14$

Now, substituting these values of $a,b,c$, and $d$ in the cubic polynomial $a{x}^3+b{x}^2+cx+d$.

Hence, the required polynomial is ${\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{7}\mathit{x}\mathbf{+}\mathbf{14}$.