Question

If -1,1 & 3 are the zeroes of a cubic polynomial, which of the following is the correct expression of the polynomial?

• A. $x^3+3x^2-x+3$
• B. $x^3-3x^2+x+3$
• C. $x^3-3x^2-x-3$
• D. $x^3-3x^2-x+3$

Answer 1

The correct option is D $x^3-3x^2-x+3$

We know that if a cubic polynomial is of the form,
$a{x}^3+b{x}^2+cx+d$.

Sum of zeroes,
$\alpha +\beta +\gamma =-\frac{b}{a}$

Sum of zeroes taken two at a time,
$\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}$

Product of zeroes,
$\alpha \beta \gamma =-\frac{d}{a}$

Assume a = 1

$-\frac{b}{a}=-b=-1+1+3=3\Rightarrow b=-3$

$\frac{c}{a}=c$

$\Rightarrow (-1)\left(1\right)+\left(1\right)\left(3\right)+\left(3\right)(-1)=c$

$\Rightarrow -1+3-3=c$

$\Rightarrow c=-1$

$-\frac{d}{a}=(-1)\left(1\right)\left(3\right)$

$\Rightarrow -d=-3$

Therefore b = -3, c = -1, d = 3

Hence, the cubic polynomial will be,

$x^3-3x^2-x+3$