Question

Find a cubic polynomial with sum of its zeroes, sum of the product of its zeroes taken two at a time and product of its zeroes as $\frac{5}{3},\frac{-11}{3}$ and $1$ respectively.

Answer 1

Find the values of $\mathit{a}\mathbf{,}\mathbf{}\mathit{b}\mathbf{,}\mathbf{}\mathit{c}\mathbf{,}\mathbf{}\mathit{d}$:

Given that,

Sum of the zeroes is $\frac{5}{3}$

Sum of the product of the zeroes taken two at a time is $\frac{-11}{3}$

Product of the zeroes is $1$

We know that for a cubic polynomial $p\left(x\right)=a{x}^3+b{x}^2+cx+d$,

Sum of its zeroes $=-\frac{b}{a}$

$=\frac{5}{3}$

Sum of the product of the zeroes taken two at a time $=\frac{c}{a}$

$=\frac{-11}{3}$

Product of the zeroes $=-\frac{d}{a}$

$$\begin{gathered} =1 \\ =\frac{3}{3} \end{gathered}$$

On solving for $a,b,c,d$ we get,

$a=3;b=-5;c=-11;d=-3$

Then the required cubic polynomial will be $p\left(x\right)=3x^3-5x^2-11x-3d$

Hence, the required cubic polynomial is $3x^3-5x^2-11x-3d$.